JEE Main · 2021 · 27 Aug 2021 · Shift 1 · Straight Lines

Let

A

be a fixed point

(0, 6)

and

B

be a moving point

(2 t, 0)

. Let

M

be the mid-point of

A B

and the perpendicular bisector of

A B

meets the

y

-axis at

C

. The locus of the mid-point

P

M C

is :

(A) $2 x^{2} + 3 y - 9 = 0$ ✓ Correct
(B) $3 x^{2} + 2 y - 6 = 0$
(C) $2 x^{2} - 3 y + 9 = 0$
(D) $3 x^{2} - 2 y - 6 = 0$

Answer: (A)

Solution

With

A (0, 6)

and

B (2 t, 0)

the midpoint is

M (t, 3)

, and

A B

has slope

- \frac{3}{t}

, so the perpendicular bisector through

M

has slope

\frac{t}{3}

y - 3 = \frac{t}{3} (x - t)

. Setting

x = 0

gives

C (0, 3 - \frac{t ^{2}}{3})

. The midpoint

P (h, k)

M C

satisfies

h = \frac{t}{2}

and

k = 3 - \frac{t ^{2}}{6}

. Eliminating

t = 2 h

k = 3 - \frac{2 h ^{2}}{3}

, so

3 k = 9 - 2 h^{2}

. The locus is

2 x^{2} + 3 y - 9 = 0

JEE Main · 2021 · 27 Aug 2021 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Straight Lines →

More JEE Main questions on Straight Lines

Let a point A lie between the parallel lines $L_{1}$ and $L_{2}$ such that its distances from $L_{1}$ and $L_{2}$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C l …
JEE Main 2026View solution →
Among the statements
(S1): If $A (5, - 1)$ and $B (- 2, 3)$ are two vertices of a triangle, whose orthocentre is $(0, 0)$ , then its third vertex is $(- 4, - 7)$ and
(S2): If positive numbers $2 a, b, c$ are three consecutive terms of an A.P., then the lines $a x + b y + c = 0$ are …
JEE Main 2026View solution →
A rectangle is formed by the lines $x = 0$ , $y = 0$ , $x = 3$ and $y = 4$ . Let the line L be perpendicular to $3 x + y + 6 = 0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $(\frac{1}{2}, - 5)$ …
JEE Main 2026View solution →
Let $A (1, 2)$ and $C (- 3, - 6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7 x - y = 14$ . If $B (α, β)$ and $D (γ, δ)$ are the other two vertices, then …
JEE Main 2026View solution →
Let $A (1, 0), B (2, - 1)$ and $C (\frac{7}{3}, \frac{4}{3})$ be three points. If the equation of the bisector of the angle ABC is $α x + β y = 5$ , then the value of $α^{2} + β^{2}$ is
JEE Main 2026View solution →
Let the angles made with the positive x-axis by two straight lines drawn from the point $P (2, 3)$ and meeting the line $x + y = 6$ at a distance $\frac{2}{3}$ from the point P be $θ_{1}$ and $θ_{2}$ . Then the value of …
JEE Main 2026View solution →