locus EAPCET STUDY MATERIAL

Chapter 1 — Locus | JR. Maths - IB
JR. MATHS  •  IB

Chapter 1  —  Locus

Synopsis  |  Exercises I – III  |  Engineering Entrance Questions (2022 – 2025)

Synopsis

  1. The path traced out by a moving point under one or more given conditions is called its “Locus”.
  2. The equation of a locus is obtained by applying the given geometrical conditions — this process is called “the equation of locus”.
  3. The locus of a point which is equidistant from the two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is the perpendicular bisector of $\overline{AB}$.
  4. The locus of a point which is at a distance $r_1$ from the given point $(x_1,y_1)$ is a circle whose equation is $$(x-x_1)^2+(y-y_1)^2=r_1^2 .$$
  5. Let $A(x_1,y_1),\ B(x_2,y_2)$ be two fixed points. The locus of $P$ such that $\angle APB = 90^\circ$ is a circle on the line joining $A,B$ as the ends of a diameter, and its equation is $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0 .$$
  6. The locus of a point $P$ such that $PA = k\,PB$ is $$(x-x_1)^2+(y-y_1)^2=k^2\!\left[(x-x_2)^2+(y-y_2)^2\right],\qquad k\neq 1 .$$ If $k=1$ it reduces to a straight line (the perpendicular bisector).
  7. If $A,B$ are two points then the locus of a point $P$ such that $PA^2+PB^2=k^2$ is
    (i) a circle if $k\neq 0$,   (ii) a straight line if $k=0$.
  8. Let $A,B$ be two points, then the locus of $P$ such that the area of $\triangle PAB$ is $A$ is a pair of parallel lines which are parallel to $AB$ and at a distance $\dfrac{2A}{AB}$ from $AB$.
  9. Let $A,B$ be two points, then the locus of a point $P$ such that $PA+PB=K$ is
    (i) an ellipse if $K>AB$,   (ii) a line segment if $K=AB$,   (iii) empty set if $K
  10. Let $A,B$ be two points, then the locus of a point $P$ such that $|PA-PB|=K$ is
    (i) a hyperbola if $Kunion of two rays if $K=AB$,   (iii) empty set if $K>AB$.
  11. If $A=(a,b),\ B=(-a,b)$ then the locus of $P$ such that $PA+PB=K$ or $PA-PB=K$ is $$\frac{4x^2}{K^2}+\frac{4(y-b)^2}{K^2-4a^2}=1 .$$
  12. If $A=(a,b),\ B=(-a,b)$ then the locus of $P$ such that $PA+PB=K$ or $PA-PB=K$ is $$\frac{4x^2}{K^2-4a^2}+\frac{4(y-b)^2}{K^2}=1 .$$
  13. The equation of the locus of a point whose distance from the $x$-axis is twice its distance from the $y$-axis is $$|y|=2|x| .$$
  14. The ends of a rod of length $K$ moves on two positive coordinate axes. The locus of the point on the rod, which divides it in the ratio $m:n$ is $$\frac{x^2}{m^2}+\frac{y^2}{n^2}=\frac{K^2}{(m+n)^2}\qquad\text{or}\qquad \frac{x^2}{n^2}+\frac{y^2}{m^2}=\frac{K^2}{(m+n)^2}.$$
  15. A straight line passing through the point $(x_1,y_1)$ meets the positive coordinate axes at $A,B$. The locus of the point $P$ which divides $AB$ in the ratio $l:m$ is $$\frac{mx_1}{x}+\frac{ny_1}{y}=m+n\qquad\text{(or)}\qquad \frac{mx_1}{x}+\frac{my_1}{y}=m+n .$$
  16. The curve represented by $$S \equiv ax^2+by^2+2hxy+2gx+2fy+c=0$$ is
    (i) a circle if $a=b,\ h=0,\ g^2+f^2-ac\geq 0$
    (ii) a pair of lines if $\Delta=0,\ h^2\geq ab,\ f^2\geq bc,\ g^2\geq ac$
    (iii) a pair of parallel lines if $\Delta=0,\ h^2=ab$
    (iv) a parabola if $\Delta\neq 0,\ h^2=ab$
    (v) an ellipse if $\Delta\neq 0,\ h^2(vi) a hyperbola if $\Delta\neq 0,\ h^2>ab$
    (vii) a rectangular hyperbola if $\Delta\neq 0,\ a+b=0$.
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Exercise – I

1. The locus of a point which is equidistant from the points $(2,3)$ and $(4,5)$ is
1) $x+y=0$
2) $x+y=7$
3) $4x+4y=38$
4) $x+y=1$
2. The locus of a point whose distance from the origin is twice its distance from the point $(1,2)$ is
1) $x^2+y^2=4\big[(x-1)^2+(y-2)^2\big]$
2) $x^2+y^2=2\big[(x-1)^2+(y-2)^2\big]$
3) $2(x^2+y^2)=(x-1)^2+(y-2)^2$
4) $x^2+y^2=(x-1)^2+(y-2)^2$
3. If $A=(-4,0)$ and $PA=2|y|$, then the locus of $P$ is
1) a circle
2) a parabola
3) an ellipse
4) a hyperbola
4. The locus of a point $P$ such that $(x-2)^2+y^2+(x-4)^2+y^2=10$ is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
5. The locus of a point $P$ which moves so that the sum of the squares of its distances from the points $(1,0)$ and $(-1,0)$ is $10$ is
1) $x^2+y^2=4$
2) $x^2+y^2=5$
3) $x^2+y^2=10$
4) $x^2+y^2=25$
6. The locus of a point which moves such that the area of the triangle formed with the points $(2,3)$ and $(-3,4)$ is $8.5$ sq. units is
1) $x^2+10xy+25y^2-34x-170y=0$
2) $x^2+10xy+25y^2-34x+170y=0$
3) $x^2-10xy+25y^2-34x-170y=0$
4) $x^2-10xy+25y^2-34x+170y=0$
7. The locus of the point which moves such that its distance from the point $(1,1)$ is twice its distance from the line $x+y+2=0$ is
1) a circle
2) a parabola
3) an ellipse
4) a hyperbola
8. The locus of a point $P$ such that the area of $\triangle PAB$ is a constant, where $A=(2,3)$ and $B=(-3,4)$, is
1) a circle
2) a pair of straight lines
3) a parabola
4) a pair of parallel straight lines
9. The locus of a point $P$ such that $PA^2+PB^2=AB^2$, where $A=(a,0)$ and $B=(-a,0)$, is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
10. Eliminating $\theta$ from $x=a\sec\theta+b\tan\theta,\ y=a\tan\theta+b\sec\theta$, the locus is
1) a circle
2) a parabola
3) a hyperbola
4) an ellipse
11. If $A=(0,5)$ and $PA=2|x|$, then the locus of $P$ is
1) a circle
2) a parabola
3) a hyperbola
4) a pair of straight lines
12. If $A=(5,-4),\ B=(7,6)$ and $3PA=2PB$, then the locus of $P$ is
1) a circle
2) a parabola
3) an ellipse
4) a hyperbola
13. The locus of a point whose distance from the $x$-axis is twice its distance from the $y$-axis is
1) $|y|=2|x|$
2) $|x|=2|y|$
3) $x=2y$
4) $y=2x$
14. The locus of a point $P$ such that $$(x+1)^2+y^2+(x-2)^2+y^2=2\big[(x-1)^2+y^2\big]$$ is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
15. If $(x-a)^2+y^2+(x+a)^2+y^2=2c^2$, then the locus of the point is
1) a circle
2) a straight line
3) an ellipse
4) a hyperbola
16. The locus of a point $P$ such that $$x^2+(y+1)^2+x^2+(y-2)^2=2\big[x^2+(y-1)^2\big]$$ is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
17. The locus of the point equidistant from the points $(a+b,\ a-b)$ and $(a-b,\ a+b)$ is
1) $bx-ay=0$
2) $bx+ay=0$
3) $ax-by=0$
4) $ax+by=0$
18. The ends of the hypotenuse of a right angled triangle are $(0,6)$ and $(-a,0)$; then the locus of the third vertex is
1) $x^2+y^2=ax$
2) $x^2+y^2=a^2$
3) $x^2+y^2+a^2=0$
4) $x^2+y^2=a^2+b^2$
19. The locus of the point $P$ such that $$\frac{x-a}{b}=\sec\theta,\qquad \frac{y-b}{a}=\tan\theta$$ (where $\theta$ is a parameter) is
1) a circle
2) a parabola
3) a hyperbola
4) an ellipse
20. If $A=(a\cos\theta,\ b\sin\theta)$, $B=(-a\sin\theta,\ b\cos\theta)$ and $O$ is the origin, $\theta$ is a parameter, then the locus of the centroid of $\triangle OAB$ is
1) $(x-a)^2+(y-b)^2=(ab)^2$
2) $\left(\dfrac{3x}{a}\right)^2+\left(\dfrac{3y}{b}\right)^2=1$
3) $\left(\dfrac{x}{a}\right)^2+\left(\dfrac{y}{b}\right)^2=1$
4) $\left(\dfrac{3x}{b}\right)^2+\left(\dfrac{3y}{a}\right)^2=1$
21. The equation of the locus of a point whose distance from the $x$-axis is equal to its distance from the $y$-axis is
1) $x^2-y^2=0$
2) $x^2+y^2=0$
3) $x^2+y^2=15$
4) $x^2-y^2=15$
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Exercise – II

22. $A(0,4),\ B(0,-4)$ are two points. The locus of $P$ which moves such that $|PA-PB|=6$ is
1) $9x^2-7y^2+63=0$
2) $9x^2-7y^2-63=0$
3) $9x^2+7y^2+63=0$
4) $9x^2+7y^2-63=0$
23. $A(2,3),\ B(-3,4)$ are two points. If a point $P$ moves such that the area of $\triangle PAB$ is $8.5$ sq. units, then the locus of $P$ is
1) $x^2+10xy+25y^2-34x-170y=0$
2) $x^2+10xy+25y^2-34x+170y=0$
3) $x^2-10xy+25y^2-34x-170y=0$
4) $x^2-10xy+25y^2-34x+170y=0$
24. A straight line segment $AB$ of length $a$ moves with its ends on the axes. The locus of the point $P$ which divides the segment in the ratio $1:2$ is
1) $9x^2+4y^2=a^2$
2) $4x^2+9y^2=a^2$
3) $9(x+y)^2=a^2$
4) $9(x+y)^2=4a^2$
25. The variable line $\dfrac{x}{a}+\dfrac{y}{b}=1$ is such that $a+b=10$. The locus of the mid-point of the portion of the line intercepted between the axes is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
26. $A=(a,0),\ B=(0,b)$ are two points. The locus of a point $P$ such that $PA+PB=AB$ is
1) a circle
2) a straight line
3) a pair of straight lines
4) an ellipse
27. The locus represented by $x^2-y^2=\dfrac{a^2}{4}\times 4$ is a
1) circle
2) parabola
3) hyperbola
4) ellipse
28. If $\dfrac{x-1}{4}=\cos\theta$ and $\dfrac{y-2}{3}=\sin\theta$, then the locus of $(x,y)$ is
1) a circle
2) a parabola
3) an ellipse
4) a hyperbola
29. The locus of the point which moves equidistant from a fixed point and a fixed straight line is
1) a circle
2) a parabola
3) an ellipse
4) a hyperbola
30. If the locus of the centroid of a triangle with vertices $(a\cos t,\ a\sin t)$, $(b\sin t,\ -b\cos t)$ and a fixed point is a circle, then the fixed point is
1) $(1,0)$
2) $(0,1)$
3) $(0,0)$
4) $(1,1)$
31. If $A=(a,0),\ B=(0,b)$, and the point $P$ divides $AB$ in the ratio $1:2$, then the locus of $P$ is
1) a circle
2) a straight line
3) a pair of straight lines
4) an ellipse
32. If $A(a,0),\ B(0,b)$ and $P$ divides $AB$ internally in the ratio $3:1$, then the locus of $P$ is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
33. If $A(a,0),\ B(0,b)$ and $P$ divides $AB$ in the ratio $3:1$, with $a^2+b^2=l^2$, then the locus of $P$ is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
34. The locus of a point whose perpendicular distances from two fixed perpendicular lines are in a constant ratio is
1) a circle
2) a pair of straight lines
3) a parabola
4) an ellipse
35. If $x=\sec\theta+\tan\theta$ and $y=\sec\theta-\tan\theta$, then the locus of $(x,y)$ is
1) a circle
2) a parabola
3) a hyperbola
4) an ellipse
36. If $\left(\dfrac{15+20\cos\theta}{5},\ \dfrac{20\sin\theta}{5}\right)=(x,y)$, then the locus of $(x,y)$ is
1) a circle
2) a parabola
3) an ellipse
4) a hyperbola
37. If $PA+PB<9$, then
1) the locus is an ellipse
2) the locus is a circle
3) the locus is a hyperbola
4) the locus is a parabola
38. $A(a_1,b_1),\ B(a_2,b_2),\ P(x,y)$ are such that $PA=PB$. Then the locus of $P$ is
1) a circle
2) a straight line
3) a parabola
4) an ellipse
39. If the sum of the distances of a point $P$ from two perpendicular lines in a plane is $1$, then the locus of $P$ is a
1) square
2) circle
3) straight line
4) pair of straight lines
40. The locus of the point $(\tan\theta+\sin\theta,\ \tan\theta-\sin\theta)$ is
1) $\left(x^2+y^2\right)^{2/3}+\left(x^2-y^2\right)^{2/3}=1$
2) $x^2-y^2=xy$
3) $x^2-y^2=12xy$
4) $\left(x^2-y^2\right)^2=16xy$
41. The curve with parametric equations $x=3(\cos t+\sin t)$, $y=4(\cos t-\sin t)$ is
1) Ellipse
2) Parabola
3) Hyperbola
4) Circle
42. A variable circle passes through the fixed point $(2,0)$ and touches the $y$-axis. Then the locus of the centre of the circle is
1) A parabola
2) A circle
3) An ellipse
4) A hyperbola
43. The curve represented by $x=2(\cos t+\sin t)$ and $y=5(\cos t-\sin t)$ is
1) Circle
2) Parabola
3) Ellipse
4) Hyperbola
44. A straight line of length $9$ units slides with its ends $A,B$ always on the $x$ and $y$ axes respectively. Locus of the centroid of $\triangle OAB$ is
1) $x^2+y^2=3$
2) $x^2+y^2=9$
3) $x^2+y^2=1$
4) $x^2+y^2=8$
45. $A(-9,0),\ B=(-1,0)$ are two points. If $P(x,y)$ is a point such that $3PB=PA$, then the locus of $P$ is
1) $x^2-y^2=9$
2) $x^2-y^2=-9$
3) $x^2+y^2=9$
4) $x^2+y^2=3$
46. $a\neq0,\ A=(a,0),\ B=(-a,0)$; locus of $P$ such that $PA^2-PB^2=4a^2$ is
1) A straight line
2) A circle
3) An ellipse
4) A parabola
47. Locus represented by $x=a(\cosh\theta+\sinh\theta)$, $y=b(\cosh\theta-\sinh\theta)$ is
1) Hyperbola
2) Parabola
3) Ellipse
4) Straight line
48. The graph represented by $x=\sin^2 t,\ y=2\cos t$ is
1) Parabola
2) Portion of parabola
3) Part of sine graph
4) Part of hyperbola
49. $A$ and $B$ are fixed points. If $PA-PB$ is a constant, locus of $P$ is
1) Parabola
2) Ellipse
3) Hyperbola
4) Circle
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Exercise – III

50. $A=(2,5),\ B=(4,-11)$ and the locus of $C$ is $9x+7y+4=0$; then the locus of the centroid of $\triangle ABC$ is
1) $27x+21y-8=0$
2) $3x+4y-2=0$
3) $24x+22y-6=0$
4) $5x+3y-7=0$
51. The algebraic sum of the perpendicular distances from the points $A(-2,0)$, $B(0,2)$ and $C(1,1)$ to a variable line be zero; then all such lines
1) Are parallel
2) Pass through a fixed point $(0,0)$
3) Form a square
4) Pass through the centroid of $\triangle ABC$
52. $A=(1,-1)$, locus of $B$ is $x^2+y^2=16$. If $P$ divides $AB$ in the ratio $3:2$ then locus of $P$ is
1) $(x-2)^2+(y-3)^2=4$
2) $(x+1)^2+(y-2)^2=4$
3) $(x-3)^2+(y-2)^2=4$
4) $(5x-2)^2+(5y+2)^2=144$
53. From a point $P$, perpendiculars $PM$, $PN$ are drawn to the $x$ and $y$ axes respectively. If $MN$ passes through the fixed point $(a,b)$, locus of $P$ is
1) $xy=ax+by$
2) $xy=ab$
3) $xy=bx+ay$
4) $x+y=xy$
54. A point moves so that the sum of the squares of its distances from the four sides of a square is constant; this point always lies on
1) straight line
2) a circle
3) parabola
4) ellipse
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Engineering Entrance Questions

AP-Engg. — 2022

1. A stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is AP-Engg. 04-07-2022 Shift-1
1) $2\pi r$
2) $\pi r^2$
3) $\tfrac12\pi r$
4) $\pi r$
2. Suppose $P$ and $Q$ are the mid points of the sides $AB$ and $BC$ of a triangle where $A(1,3)$, $B(3,7)$ and $C(7,15)$ are vertices. Then the locus of $R$ satisfying $AC^2+QR^2=PR^2$ is AP-Engg. 04-07-2022 Shift-2
1) $6x+12y=297$
2) $6x+12y+297=0$
3) $12x+6y=297$
4) $12x+6y+297=0$
3. A point $P(x,y)$ is such that its distances from $(-1,0)$ and $(0,2)$ are in a ratio of $\sqrt2:1$. Then the locus of $P$ is AP-Engg. 06-07-2022 Shift-1
1) $(x-1)^2+(y-4)^2=10$
2) $(x+2)^2+(y+2)^2=10$
3) $(x-1)^2+(y-4)^2=100$
4) $(x+2)^2+(y+2)^2=100$
4. A variable line passing through $(l,m)$ intersects the coordinate axes at the points $A$ and $B$. If the lines drawn parallel to the $y$-axis through $A$ and parallel to the $x$-axis through $B$ meet at $P$, then the locus of $P$ is AP-Engg. 06-07-2022 Shift-2
1) $\dfrac{l}{x}+\dfrac{m}{y}=1$
2) $\dfrac{x}{l}+\dfrac{y}{m}=1$
3) $\dfrac{m}{x}+\dfrac{l}{y}=1$
4) $\dfrac{x}{m}+\dfrac{y}{l}=1$
5. On the locus of the point $P(x,y)$ equidistant from $(3,0)$ and $(0,4)$, if $A$ and $B$ are two points that satisfy $4x=3y$ and $x=y$ respectively, then the distance between $A$ and $B$ is AP-Engg. 07-07-2022 Shift-1
1) $\dfrac52$
2) $5$
3) $25$
4) $\dfrac{25}{2}$
6. A point $P(x,y)$ is such that the sum of squares of its distances from $(a,0)$ and $(-a,0)$ is $2b^2$. The equation representing the locus of $P$ is AP-Engg. 07-07-2022 Shift-2
1) $x^2+y^2=b^2+a^2$
2) $x^2+y^2=b^2-a^2$
3) $x^2+y^2=b^2-2a^2$
4) $x^2+y^2=b^2+2a^2$
7. If a line $AB$ of length $r$ moves so that $A$ and $B$ always lie respectively on the $x$-axis and $y=6x$, then the locus of the midpoint of $AB$ is AP-Engg. 08-07-2022 Shift-2
1) $y=12x$
2) $\left(x-\dfrac{y}{3}\right)^2+y^2=\dfrac{r^2}{4}$
3) $y=\dfrac{x}{6}$
4) $y=6x$

TS-EAMCET — 2022

8. Let $A(5,-3)$, $B(3,-2)$, $C(-1,5)$ be three points. If $P$ is a point satisfying the condition $PA^2+2PB^2=3PC^2$, then a point that lies on the locus of $P$ is TS-Engg. 18-07-2022 Shift-1
1) $\left(-\dfrac17,\ \dfrac12\right)$
2) $\left(-\dfrac52,\ -2\right)$
3) $\left(-\dfrac{2}{21},\ \dfrac{31}{66}\right)$
4) $\left(2,\ \dfrac{37}{22}\right)$
9. If the perimeter of a triangle is $20$ and two of its vertices are $(-5,0)$ and $(6,0)$, then the locus of the third vertex is TS-Engg. 18-07-2022 Shift-2
1) $40x^2-81y^2-40x-800=0$
2) $40x^2+9y^2-25x+800=0$
3) $40x^2-9y^2=800$
4) $5y^2-3y^2+3x-4y+25=0$
10. If $A(1,1)$, $B(-1,1)$ and $C(-1,-1)$ are three points and a point $P$ moves such that $PA^2=PB^2+PC^2$, then the equation of the locus of $P$ is TS-Engg. 19-07-2022 Shift-2
1) $x^2+y^2-6x-2y+2=0$
2) $x^2+y^2+6x+2y+2=0$
3) $x^2+y^2+6x-2y+2=0$
4) $x^2+y^2+6x+2y-2=0$
11. $B(2,3)$, $C(5,-2)$, $D(1,-1)$ are three points. If $A$ is a variable point such that the area of the quadrilateral $ABCD$ is $10$ sq. units, then the locus of $A$ is TS-Engg. 20-07-2022 Shift-1
1) $(x-4y+42)(x-4y+2)=0$
2) $(x-4y-42)(x-4y-2)=0$
3) $(4x-y+42)(4x-y+2)=0$
4) $(4x-y-42)(4x-y-2)=0$

AP-EAPCET — 2023

12. The locus of a point which is at a distance of $2$ units from the line $2x-3y+4=0$ and at a distance of $\sqrt{13}$ units from the point $(5,0)$ is AP-Engg. 15-05-2023 Shift-1
1) $8x^2+12xy+56x-24y+84=0$
2) $12xy-5y^2-56x+24y+84=0$
3) $8x^2+12xy+y^2-56x+24y+84=0$
4) $8x^2+12xy-7y^2-56x+24y+84=0$
13. If $A(4,0)$ and $B(-4,0)$ are two points, then the locus of a point $P$ such that $PA-PB=4$ is AP-Engg. 15-05-2023 Shift-2
1) $3x^2-y^2=12$
2) $x^2-3y^2=12$
3) $4\left(x^2-3y^2\right)=1$
4) $3x^2-y^2=1$
14. If $A=(2,3)$ and $B=(-4,5)$ are two fixed points, then the locus of a point $P$ such that the area of $\triangle PAB$ is $12$ square units is AP-Engg. 16-05-2023 Shift-1
1) $x^2+6xy+9y^2+22x+66y+23=0$
2) $x^2-6xy+9y^2+22x+66y+23=0$
3) $x^2+6xy+9y^2-22x-66y-23=0$
4) $x^2-6xy+9y^2-22x-66y-23=0$
15. The locus of the point which is equidistant from the point $(1,1)$ and the line $x+y+1=0$ is AP-Engg. 17-05-2023 Shift-1
1) $x^2-y^2+6x+4y-3=0$
2) $(x-y)^2-6(x+y)+3=0$
3) $(x+y)^2+6(x-y)+3=0$
4) $x^2+y^2-2x-2y+4=0$
16. If $t\in\mathbb{R}-\{-1\}$, then the locus of the point $\left(\dfrac{3at}{1+t^3},\ \dfrac{3at^2}{1+t^3}\right)$ is AP-Engg. 17-05-2023 Shift-2
1) $x^3+y^3=3ax^2y^2$
2) $x^3+y^3=3axy$
3) $x^2+y^2=3axy$
4) $x^2+y^2=3a^2xy$
17. The locus of a point which moves such that its distance from the origin is three times its distance from the point $(1,2)$ is
1) $x^2+y^2=9\big[(x-1)^2+(y-2)^2\big]$
2) $3(x^2+y^2)=(x-1)^2+(y-2)^2$
3) $x^2+y^2=3\big[(x-1)^2+(y-2)^2\big]$
4) $9(x^2+y^2)=(x-1)^2+(y-2)^2$
18. The locus of a point $P$ such that the sum of its distances from the points $(2,0)$ and $(-2,0)$ is $6$ is
1) a circle
2) an ellipse
3) a hyperbola
4) a parabola

TS-EAMCET — 2023

19. If $t$ is a parameter, $A=(a\sec t,\ b\tan t)$, $B=(-a\tan t,\ b\sec t)$ and $O=(0,0)$, then the locus of the centroid of $\triangle OAB$ is TS-Engg. 12-05-2023 Shift-1
1) $9xy=ab$
2) $x^2-9y^2=a^2-b^2$
3) $xy=9ab$
4) $x^2-y^2=\dfrac12\left(a^2-b^2\right)$
20. The locus of the mid points of the intercepted portion of the tangents by the coordinate axes, which are drawn to the ellipse $x^2+2y^2=2$, is TS-Engg. 12-05-2023 Shift-2
1) $\dfrac{1}{2x^2}+\dfrac{1}{4y^2}=1$
2) $\dfrac{1}{4x^2}+\dfrac{1}{2y^2}=1$
3) $\dfrac{x^2}{2}+\dfrac{y^2}{4}=1$
4) $\dfrac{x^2}{4}+\dfrac{y^2}{2}=1$
21. Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distances of $P$ from $A$ and $B$ is $4$. Then the equation of the locus of the point $P$ is TS-Engg. 12-05-2023 Shift-2
1) $3x^2-2xy+3y^2-4x+12y+16=0$
2) $3x^2-2xy+3y^2-8x+8y=0$
3) $3x^2+2xy+3y^2+8x-8y=0$
4) $3x^2+2xy+3y^2+4x-12y+16=0$
22. Let $A=(1,2)$, $B=(2,1)$, $C=(-1,-1)$ be three points. If $P$ is a point such that the area of the quadrilateral $PABC$ is twice the area of the triangle $PAB$, then the equation of the locus of $P$ is TS-Engg. 13-05-2023 Shift-2
1) $8x^2-14xy+3y^2-18x+22y+7=0$
2) $9x^2-12xy+4y^2-24x+16y+16=0$
3) $x^2+2xy+y^2-6x-6y+9=0$
4) $x^2-4xy+8y^2-4=0$
23. If a point $P$ moves so that the distance from $(0,2)$ to $P$ is $\dfrac{1}{\sqrt2}$ times the distance of $P$ from $(-1,0)$, then the locus of the point $P$ is TS-Engg. 14-05-2023 Shift-1
1) a circle with centre $(1,4)$ and radius $10$ units
2) a circle with centre $(-1,-4)$ and radius $\sqrt{10}$ units
3) a circle with centre $(1,4)$ and radius $\sqrt{10}$ units
4) a parabola with focus at $(1,4)$ and length of latus rectum $10$ units
24. A straight line passing through a fixed point $(-3,4)$ intersects the coordinate axes at $A$ and $B$. If $O$ is the origin and $OABC$ forms a rectangle, then the locus of $C$ is TS-Engg. 14-05-2023 Shift-2
1) $xy+3x-4y=0$
2) $xy-3x+4y=0$
3) $xy-3x-4y=0$
4) $xy+3x+4y=0$

AP-EAPCET — 2024

25. The equation $axy+byz=cy$ represents the locus of the points which lie on AP-Engg. 18-05-2024 Shift-1
1) $zx$-plane or on the planes perpendicular to $zx$-plane
2) on the planes perpendicular to $x$-axis
3) on the lines perpendicular to $zx$-plane
4) on the lines perpendicular to $xy$-plane
26. If a variable straight line passing through the point of intersection of the lines $x-2y+3=0$ and $2x-y-1=0$ intersects the $X$ and $Y$ axes at $A$ and $B$ respectively, then the equation of the locus of a point which divides the segment $AB$ in the ratio $-2:3$ is
1) $14x^2+3xy-15y^2=0$
2) $xy=14x+15y$
3) $x^2+xy-y^2=0$
4) $14x+3xy-15y=0$
27. If the line segment joining the points $(1,0)$ and $(0,1)$ subtends an angle of $45^\circ$ at a variable point $P$, then the equation of the locus of $P$ is AP-Engg. 20-05-2024 Shift-1
1) $(x^2+y^2-1)(x^2+y^2-2x-2y+1)=0,\ x\neq0,1$
2) $(x^2+y^2-1)(x^2+y^2+2x+2y+1)=0,\ x\neq0,1$
3) $x^2+y^2+2x+2y+1=0$
4) $x^2+y^2=4$
28. $A(2,3)$, $B(-1,1)$ are two points. If $P$ is a variable point such that $\angle APB=90^\circ$, then locus of $P$ is AP-Engg. 20-05-2024 Shift-1
1) $x^2+y^2-x-4y+1=0$
2) $x^2+y^2+x+4y-1=0$
3) $x^2+y^2-x+4y-1=0$
4) $x^2+y^2+x-4y+1=0$
29. $P$ is a variable point such that the distance of $P$ from $A(4,0)$ is twice the distance of $P$ from $B(-4,0)$. If the line $3y-3x-20=0$ intersects the locus of $P$ at the points $C$ and $D$, then the distance between $C$ and $D$ is AP-Engg. 21-05-2024 Shift-1
1) $8$
2) $\dfrac{8\sqrt2}{3}$
3) $\dfrac{32}{3}$
4) $\dfrac{8}{3}$
30. The perimeter of the locus of the point $P$ which divides the line segment $QA$ internally in the ratio $1:2$, where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$, is AP-Engg. 21-05-2024 Shift-2
1) $8\pi$
2) $4\pi$
3) $\pi$
4) $9\pi$
31. The equation of the locus of points which are equidistant from the points $(2,3)$ and $(4,5)$ is AP-Engg. 22-05-2024 Shift-1
1) $x+y=0$
2) $x+y=7$
3) $4x+4y=38$
4) $x+y=1$
32. The locus of a variable point which forms a triangle of fixed area with two fixed points is AP-Engg. 22-05-2024 Shift-2
1) a circle
2) a circle with fixed points as ends of a diameter
3) a pair of non parallel lines
4) a pair of parallel lines
33. The locus of the midpoint of the portion of the line $x\cos\alpha+y\sin\alpha=p$ intercepted by the coordinate axes, where $p$ is a constant, is
1) $\dfrac{1}{x^2}+\dfrac{1}{y^2}=\dfrac{3}{p^2}$
2) $\dfrac{1}{x^2}+\dfrac{1}{y^2}=-\dfrac{1}{p^2}$
3) $x^2+y^2=2p^2$
4) $\dfrac{2}{x^2}+\dfrac{2}{y^2}=\dfrac{1}{p^2}$

TG-EAMCET — 2024

34. The centroid of a variable triangle $ABC$ is at a distance of $5$ units from the origin. If $A=(2,3)$ and $B=(3,2)$ then the locus of $C$ is TG-Engg. 09-05-2024 Shift-1
1) a circle of radius $225$ units
2) a rectangular hyperbola
3) a circle of diameter $30$ units
4) an ellipse with eccentricity $\dfrac45$
35. If the ratio of the distance of a variable point $P$ from the point $(1,1)$ and the line $x-y+2=0$ is $1:\sqrt2$ then the equation of the locus of $P$ is TG-Engg. 09-05-2024 Shift-2
1) $x^2+2xy+y^2-8x=0$
2) $x^2+2xy+3y^2-12x-4y+4=0$
3) $x^2+2xy+y^2-12x+4y+4=0$
4) $x^2+2xy+y^2-8x+8y=0$
36. If the distance from a variable point $P$ to the point $(4,3)$ is equal to the perpendicular distance from $P$ to the line $x+2y-1=0$, then the equation of the locus of the point $P$ is TG-Engg. 10-05-2024 Shift-1
1) $4x^2+4xy+y^2-38x+26y+124=0$
2) $4x^2-4xy+y^2-38x-26y+124=0$
3) $4x^2-4xy+y^2+38x+26y+124=0$
4) $4x^2-4xy+y^2-38x+26y+124=0$
37. If the locus of the centroid of the triangle with vertices $A(a,0)$, $B(a\cos t,\ a\sin t)$ and $C(b\sin t,\ -b\cos t)$ ($t$ is a parameter) is $9x^2+9y^2-6x=49$, then the area of the triangle formed by the line $\dfrac{x}{a}+\dfrac{y}{b}=1$ with the coordinate axes is TG-Engg. 10-05-2024 Shift-2
1) $\dfrac{49}{2}$
2) $\dfrac{7}{2}$
3) $\dfrac{1}{2}$
4) $\dfrac{47}{2}$
38. $P$ and $Q$ are the points of trisection of the line segment joining the points $(3,-7)$ and $(-5,3)$. If $PQ$ subtends a right angle at a variable point $R$, then the locus of $R$ is TG-Engg. 11-05-2024 Shift-1
1) a circle with radius $\dfrac{\sqrt{41}}{3}$
2) a circle with radius $\sqrt{409}$
3) a pair of straight lines passing through $(-1,-2)$
4) a pair of straight lines passing through $(1,2)$

AP-EAPCET — 2025

39. If the distance of a variable point $P$ from a point $A(2,-2)$ is twice the distance of $P$ from the $Y$-axis, then the equation of locus of $P$ is AP-Engg. 21-05-2025 Shift-1
1) $3x^2-y^2+4x-4y-8=0$
2) $x^2-4x+4y+8=0$
3) $3x^2-y^2+4x-4y+8=0$
4) $y^2-4x+4y+8=0$
40. If $P$ is a variable point which is at a distance of $2$ units from the line $2x-3y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is AP-Engg. 21-05-2025 Shift-2
1) $4x^2+12xy-5y^2-44x-42y+245=0$
2) $12xy-5y^2-44x-42y+243=0$
3) $8x^2+12xy-5y^2-44x-42y+243=0$
4) $12xy-13y^2-44x-42y+245=0$
41. A straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $OPRQ$ is a rectangle, then the locus of $R$ is AP-Engg. 22-05-2025 Shift-1
1) $3x+2y=xy$
2) $2x+3y=xy$
3) $3x+2y=6$
4) $2x+3y=6$
42. If $A(1,0)$, $B(0,-2)$, $C(2,-1)$ are three fixed points, then the equation of the locus of a point $P$ such that area of $\triangle PAB$ is equal to area of $\triangle PAC$ is AP-Engg. 22-05-2025 Shift-2
1) $x^2-2xy-2y^2+2x-2y+1=0$
2) $x^2-2xy+2y^2-2x+2y+1=0$
3) $x^2-2xy-2x+2y+1=0$
4) $x^2-2xy+2x-2y+1=0$
43. Let $A(5,4)$ and $B(5,-4)$ be two points. If $P$ is a point in the coordinate plane such that $\angle APB=\dfrac{\pi}{4}$, then the point $P$ lies on the curve AP-Engg. 23-05-2025 Shift-1
1) $x^2+y^2+10x-17=0$
2) $x^2+y^2-2x-31=0$
3) $x^2+y^2-10x+17=0$
4) $x^2+y^2+2x-31=0$
44. If the locus of a point which is equidistant from the coordinate axes forms a triangle with the line $y=3$, then the area of the triangle is AP-Engg. 23-05-2025 Shift-2
1) $18$
2) $9$
3) $6$
4) $3$
45. $A(a,0)$ is a fixed point and $\theta$ is a parameter such that $0<\theta<2\pi$. If $P(a\cos\theta,\ a\sin\theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b\sin\theta,\ -b\cos\theta)$ is a point on the circle $x^2+y^2=b^2$, then the locus of the centroid of the triangle $APQ$ is AP-Engg. 24-05-2025 Shift-1
1) A circle with centre at $\left(\dfrac{a}{3},0\right)$ and radius $\dfrac{\sqrt{a^2+b^2}}{3}$
2) A circle with centre at $(a,0)$ and radius $\dfrac{\sqrt{a^2+b^2}}{3}$
3) A parabola with focus at $\left(\dfrac{a}{3},0\right)$
4) A parabola with focus at $(a,0)$
46. $A(4,3)$, $B(2,5)$ are two points. If $P$ is a variable point on the same side as that of the origin with respect to the line $AB$ and is at most at a distance of $5$ units from the midpoint of $AB$, then the locus of $P$ is AP-Engg. 26-05-2025 Shift-1
1) $x^2+y^2-6x-8y=0$
2) $x^2+y^2-6x-8y\leq0,\ x+y-7<0$
3) $x^2+y^2+6x+8y-25=0,\ x+y-7\geq0$
4) $x^2+y^2-6x+8y\geq0,\ x+y-7<0$
47. If $A(\cos\alpha,\sin\alpha)$, $B(\sin\alpha,-\cos\alpha)$, $C(1,2)$ are the vertices of a $\triangle ABC$ then the locus of its centroid is AP-Engg. 26-05-2025 Shift-2
1) $3(x^2+y^2)-2x-4y+1=0$
2) $x^2+y^2-2x-4y+1=0$
3) $x^2+y^2-2x-4y+3=0$
4) $2(x^2+y^2)-2x-4y+5=0$
48. The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are $(1,2)$ and $(4,5)$, is AP-Engg. 27-05-2025 Shift-1
1) $x^2+y^2+5x+7y+14=0$
2) $3x+3y-1=0$
3) $3x+3y+1=0$
4) $x^2+y^2-5x-7y+14=0$

TS-EAPCET — 2025

49. If the points $A(2,3)$, $B(3,2)$ form a triangle with a variable point $(t,t^2)$, where $t$ is a parameter, then the equation of the locus of the centroid of triangle $ABC$ is TS-Engg. 02-05-2025 Shift-1
1) $9x^2-30x-3y+20=0$
2) $3x^2-10x-y+10=0$
3) $9y^2-30y-3x+20=0$
4) $3y^2-10y-x+10=0$
50. A line segment joining a point $A$ on the $x$-axis to a point $B$ on the $y$-axis is such that $AB=15$. If $P$ is a point on $AB$ such that $\dfrac{AP}{PB}=\dfrac{2}{3}$ then the locus of $P$ is TS-Engg. 02-05-2025 Shift-2
1) $x=9\cos\theta,\ y=6\sin\theta$
2) $x=6\cos\theta,\ y=9\sin\theta$
3) $x=6\cos\theta,\ y=6\sin\theta$
4) $x=9\cos\theta,\ y=9\sin\theta$
51. The equation of the locus of a point which is at a distance of $5$ units from a fixed point $(1,4)$ and also from a fixed line $2x+3y-1=0$ is TS-Engg. 03-05-2025 Shift-1
1) $9x^2+12xy+4y^2-30x-108y+222=0$
2) $9x^2-12xy+4y^2-30x-98y+220=0$
3) $9x^2+12xy+4y^2-22x-108y+222=0$
4) $9x^2-12xy+4y^2-22x-98y+220=0$
52. A straight line passing through a point $(3,2)$ cuts the $X$ and $Y$ axes at the points $A$ and $B$ respectively. If a point $P$ divides $AB$ in the ratio $2:3$, then the equation of the locus of point $P$ is TS-Engg. 03-05-2025 Shift-2
1) $\dfrac{9}{x}+\dfrac{4}{y}=1$
2) $9x+4y=5xy$
3) $4x+9y=5xy$
4) $\dfrac{4}{x}+\dfrac{9}{y}=1$
53. If $A=(0,1)$, $B=(1,2)$, $C=(-2,1)$ then the equation of the locus of a point $P$ such that area of triangle $PAB$ = area of triangle $PAC$ is TS-Engg. 04-05-2025 Shift-1
1) $x^2-2xy-3y^2+2x+6y-3=0$
2) $x^2+2xy-3y^2+2x+6y-4=0$
3) $x^2-2xy-3y^2+2x-6y+4=0$
4) $x^2-2xy+3y^2-2x+6y-3=0$
54. $A(2,0)$, $B(0,2)$, $C(-2,0)$ are three points. Let $a,b,c$ be the perpendicular distances from a variable point $P$ on to the lines $AB$, $BC$ and $CA$ respectively. If $a,b,c$ are in arithmetic progression, then the locus of $P$ is TS-Engg. 04-05-2025 Shift-2
1) $\sqrt2\,y=2|x-y+2|-|x+y-2|$
2) $\sqrt2\,|y|=|x-y+2|-|x+y-2|$
3) $2|x-y+2|=\left|\dfrac{x+y-2}{\sqrt2}\right|+\left|\dfrac{x-y-2}{\sqrt2}\right|$
4) $2|x-y+2|=\left|x+\left(\sqrt2+1\right)y+2\right|$
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Answer Keys

Exercise – I
Q1234567891011
Ans 11423 444123
Q12131415161718192021
Ans 21211 42314
Exercise – II
Q2223242526272829303132333435
Ans 11233 22212 1321
Q3637383940414243444546474849
Ans 31114 11323 1113
Exercise – III
Q5051525354
Ans 14432
Engineering Entrance Questions
Q1234567891011121314
Ans 41111 23412 4213
Q1516171819202122232425262728
Ans 23111 12433 1411
Q2930313233343536373839404142
Ans 32242 32221 1213
Q434445464748495051525354
Ans 22121 42143 11
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Hints & Solutions

Exercise – I

  1. The perpendicular distance from $P(x_1,y_1)$ to the $x$-axis is $|y|$ and to the $y$-axis is $|x|$.
  2. $x^2+y^2=4\left[(x-1)^2+(y-2)^2\right]$
  3. $A=(-4,0)$,   $PA=2|y|$
  4. $(x-2)^2+y^2+(x-4)^2+y^2=10$
  5. Use the area of the triangle formula.
  6. Area of the triangle is $0$.
  7. $AC^2+BC^2=AB^2$
  8. Eliminate $\theta$.
  9. $(x,y)=\left(\dfrac{a\sec t-a\tan t}{3},\ \dfrac{b\tan t+b\sec t}{3}\right)$; eliminate $t$.
  10. Eliminate $\theta$.
  11. $A(0,5);\ PA=2|x|$
  12. $A(5,-4),\ B(7,6)$;   $3PA=2PB$
  13. $|y|=2|x|$
  14. $(x+1)^2+y^2+(x-2)^2+y^2=2\left[(x-1)^2+y^2\right]$
  15. $(x-a)^2+y^2+(x+a)^2+y^2=2c^2$
  16. $x^2+(y+1)^2+x^2+(y-2)^2=2\left[x^2+(y-1)^2\right]$
  17. Locus is the perpendicular bisector.
  18. $A=(a,0),\ B=(-a,0),\ P(x,y)$;   $AP^2+BP^2=AB^2$
  19. $\dfrac{x-a}{b}=\sec\theta,\quad \dfrac{y-b}{a}=\tan\theta$
  20. $(x,y)=\left(\dfrac{a\cos\theta+a\sin\theta}{3},\ \dfrac{b\sin\theta+b\cos\theta}{3}\right)$; eliminate $\theta$.
  21. $|x|^2+|y|^2=15$

Exercise – II

  1. $PA=PB\pm6$
  2. Use the area of the triangle formula.
  3. $A(p,0),\ B(0,q)$; use the section formula.
  4. $P(x,y)=\left(\dfrac{a}{2},\dfrac{b}{2}\right)$
  5. $PA+PB=AB \Rightarrow P,A,B$ are collinear.
  6. $x^2-y^2=\dfrac{a^2}{4}\times4$
  7. $\dfrac{x-1}{4}=\cos\theta,\ \dfrac{y-2}{3}=\sin\theta$ and $\cos^2\theta+\sin^2\theta=1$.
  8. The locus of a point which moves equidistant from a fixed point and a fixed straight line is a parabola.
  9. $P(x,y)=\left(\dfrac{a\cos t+b\sin t+1}{3},\ \dfrac{a\sin t-b\cos t}{3}\right)$
  10. $\dfrac{4x^2}{k^2}+\dfrac{4(y-b)^2}{k^2-4a^2}=1$
  11. $A(a,0),\ B(0,b)$;   $P(x,y)=\left(\dfrac{a}{2},\dfrac{b}{2}\right)$
  12. $a^2+b^2=r^2$
  13. $A(a,0),\ B(0,b)$;   $P(x,y)=\left(\dfrac{3a}{4},\dfrac{b}{4}\right)$
  14. $a^2+b^2=l^2$
  15. Consider the perpendicular lines as the coordinate axes.
  16. $x=\sec\theta+\tan\theta,\quad y=\sec\theta-\tan\theta$
  17. $\left(\dfrac{15+20\cos\theta}{5},\ \dfrac{20\sin\theta}{5}\right)=(x,y)$
  18. $PA+PB<9$
  19. $A(a_1,b_1),\ B(a_2,b_2),\ P(x,y);\quad PA=PB$
  20. $|x|+|y|=1$
  21. Eliminate $\theta$.
  22. $\dfrac{x}{3}=\cos t+\sin t,\quad \dfrac{y}{4}=\cos t-\sin t$
  23. The locus of a point which moves equidistant from a fixed point and a fixed straight line is a parabola.
  24. $\dfrac{x}{2}=\cos t+\sin t,\quad \dfrac{y}{5}=\cos t-\sin t$
  25. $P(x,y)=\left(\dfrac{a}{3},\dfrac{b}{3}\right)$
  26. $a^2+b^2=9$
  27. $9PB^2=PA^2$
  28. $(x-a)^2+y^2-\left[(x+a)^2+y^2\right]=4a^2$
  29. $\dfrac{x}{a}=\cosh\theta+\sinh\theta,\quad \dfrac{y}{b}=\cosh\theta-\sinh\theta$
  30. $x=1-\cos^2 t,\quad \dfrac{y}{2}=\cos t$

Exercise – III

  1. Standard result.
  2. Let $C'(\alpha,\beta)$; $(x,y)=\left(\dfrac{6+\alpha}{3},\dfrac{-6+\beta}{3}\right)$ so that $(\alpha,\beta)=(3x-6,\ 3y+6)$; substitute in $9x+7y+4=0$.
  3. Algebraic sum of the perpendicular distances from three non-collinear points is $0$; then the line passes through the centroid of the triangle formed by these points.
  4. $A(1,-1),\ B(\alpha,\beta),\ P(x,y)$;   $(x,y)=\left(\dfrac{3\alpha+2}{5},\dfrac{3\beta-2}{5}\right)$; find $\alpha,\beta$ and substitute in $x^2+y^2=16$.
  5. Let $P(\alpha,\beta)$. The equation of the line passing through $M,N$ is $\beta x+\alpha y=\alpha\beta$; it passes through $(a,b)$.
  6. Take the four sides of the square as $(0,0),(a,0),(a,a),(0,a)$.

Entrance Questions — Selected Hints

  1. Midpoint of the stick traces a circle of radius $r/2$; its circumference is $\pi r$.
  2. Use the mid-point formula and the distance formula.
  3. Use $PA^2=2\,PB^2$ and simplify.
  4. Use the intercept form and the condition that the intercepts are $a$ and $b$ with $\dfrac{l}{a}+\dfrac{m}{b}=1$.
  5. First find the locus (a straight line), then find $A$ and $B$ and hence $AB$.
  6. $PA^2+PB^2=2b^2 \Rightarrow x^2+y^2=b^2-a^2$.
  7. Use the midpoint formula with $A$ on the $x$-axis and $B$ on $y=6x$.
  8. Substitute the coordinates of each option in the locus.
  9. Perimeter $=20$ gives $PA+PB=9$; an ellipse.
  10. Use $PA^2=PB^2+PC^2$ and simplify.
  11. Use the area of the quadrilateral as the sum of two triangles.
  12. Use the distance-from-a-line and distance-from-a-point conditions simultaneously.
  13. Use the standard hyperbola result $PA-PB=2a$.
  14. Area of a triangle with a fixed base = constant gives a pair of parallel lines.
  15. Use the focus–directrix definition (parabola).
  16. Eliminate $t$ using $x^3+y^3=3axy$ (the folium of Descartes).
  17. Use $PO^2=9\,PA^2$.
  18. Sum of distances $=6>4$; hence an ellipse.
  19. Centroid $=\left(\dfrac{a\sec t-a\tan t}{3},\dfrac{b\tan t+b\sec t}{3}\right)$; eliminate $t$.
  20. Use the standard tangent to an ellipse and the mid-point formula.
  21. Sum of distances from two fixed points $=4$; an ellipse.
  22. Use the area condition with the quadrilateral split into two triangles.
  23. Apply $PA^2=\dfrac12\,PB^2$ and simplify to a circle.
  24. Use the intercept form and the rectangle condition.
  25. Interpret the equation as a plane in 3-D.
  26. Use the section formula with the given ratio and eliminate the parameter.
  27. Use the condition that a chord subtends $45^\circ$ at a point on a circle.
  28. Use the diameter form: $(x-2)(x+1)+(y-3)(y-1)=0$.
  29. Use $PA=2\,PB$ to get the circle, then find its intersection with the given line.
  30. Use the section formula; the locus is a circle of radius $3$ times the original, so perimeter $=8\pi$.
  31. Perpendicular bisector of the join of $(2,3)$ and $(4,5)$.
  32. Fixed area with a fixed base gives a pair of parallel lines.
  33. Midpoint $=\left(\dfrac{p}{2\cos\alpha},\dfrac{p}{2\sin\alpha}\right)$; eliminate $\alpha$.
  34. Centroid is at distance $5$ from the origin; the locus is a circle of radius $15$ (diameter $30$).
  35. Use the distance-ratio condition and simplify.
  36. Use the focus–directrix definition (parabola).
  37. Eliminate the parameter $t$ from the centroid coordinates.
  38. Use the right-angle condition $PR^2+QR^2=PQ^2$.
  39. Use $PA=2|x|$ and simplify.
  40. Use the distance-from-a-line and distance-from-a-point conditions.
  41. Use the intercept form with the rectangle condition $x=a,\ y=b$.
  42. Equal areas give a pair of straight lines through the common vertex.
  43. Use the condition that a chord subtends a given angle at a point on a circle.
  44. The locus is $|x|=|y|$; intersect with $y=3$.
  45. Centroid $=\left(\dfrac{a+a\cos\theta+b\sin\theta}{3},\dfrac{a\sin\theta-b\cos\theta}{3}\right)$; eliminate $\theta$.
  46. Use the inequality describing a disc and a half-plane.
  47. Centroid $=\left(\dfrac{\cos\alpha+\sin\alpha+1}{3},\dfrac{\sin\alpha-\cos\alpha+2}{3}\right)$; eliminate $\alpha$.
  48. Use the diameter form for the circle.
  49. Eliminate $t$ from the centroid coordinates.
  50. Use the section formula with $AP:PB=2:3$ and $a^2+b^2=225$.
  51. Equate the two distance expressions.
  52. Use the section formula with the ratio $2:3$.
  53. Equal areas give a pair of straight lines.
  54. Use the condition that $a,b,c$ are in A.P.: $2b=a+c$.
Note: This HTML page is a clean, re-typeset version of the scanned chapter. A few options that were illegible or cut off in the original scan have been reconstructed from context or marked accordingly. Always cross-check with your textbook / official answer key.
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