Locus and its equation — practice questions

Question 1 · difficulty L2 · understanding

If two tangents drawn from a point P to the parabola y 2 = 16(x $$-$$ 3) are at right angles, then the locus of point P is :

• A. x + 3 = 0
• B. x + 1 = 0
• C. x + 2 = 0
• D. x + 4 = 0

Question 2 · difficulty L3 · understanding

Let A be the point (3, 0) and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is $e$, then $72e^2$ is equal to ________.

Question 3 · difficulty L3 · understanding

The locus of the point of intersection of the lines $$\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$$ and $$\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0$$ is a conic, whose eccentricity is _________.

Question 4 · difficulty L3 · understanding

If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta PAB$ lies on the line :

• A. $x + 9y = 36$
• B. $9x - 9y = 32$
• C. $4x - 9y = 12$
• D. $6x - 9y = 20$

Question 5 · difficulty L3 · understanding

If the locus of the point, whose distances from the point $$(2,1)$$ and $$(1,3)$$ are in the ratio $$5: 4$$, is $$a x^2+b y^2+c x y+d x+e y+170=0$$, then the value of $$a^2+2 b+3 c+4 d+e$$ is equal to :

• A. 37
• B. $$-27$$
• C. 437
• D. 5

Question 6 · difficulty L3 · understanding

Let $$\mathrm{A}(-1,1)$$ and $$\mathrm{B}(2,3)$$ be two points and $$\mathrm{P}$$ be a variable point above the line $$\mathrm{AB}$$ such that the area of $$\triangle \mathrm{PAB}$$ is 10. If the locus of $$\mathrm{P}$$ is $$\mathrm{a} x+\mathrm{by}=15$$, then $$5 \mathrm{a}+2 \mathrm{~b}$$ is :

• A. $$-\frac{12}{5}$$
• B. $$-\frac{6}{5}$$
• C. 6
• D. 4

Question 7 · difficulty L3 · understanding

If the point $$\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$$ lies on the curve traced by the mid-points of the line segments of the lines $$x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$$ between the co-ordinates axes, then $$\alpha$$ is equal to :

• A. $$-$$7
• B. 7
• C. $$-$$7$$\sqrt3$$
• D. 7$$\sqrt3$$

Question 8 · difficulty L3 · understanding

A point $$P$$ moves so that the sum of squares of its distances from the points $$(1,2)$$ and $$(-2,1)$$ is 14. Let $$f(x, y)=0$$ be the locus of $$\mathrm{P}$$, which intersects the $$x$$-axis at the points $$\mathrm{A}$$, $$\mathrm{B}$$ and the $$y$$-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to :

• A. $${9 \over 2}$$
• B. $${{3\sqrt {17} } \over 2}$$
• C. $${{3\sqrt {17} } \over 4}$$
• D. 9

Question 9 · difficulty L3 · understanding

Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

• A. 3x 2 $$-$$ 2y $$-$$ 6 = 0
• B. 3x 2 + 2y $$-$$ 6 = 0
• C. 2x 2 + 3y $$-$$ 9 = 0
• D. 2x 2 $$-$$ 3y + 9 = 0

Question 10 · difficulty L3 · understanding

Let O be the vertex of the parabola $y^2=4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :

• A. 1
• B. 2
• C. 4
• D. 8