JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Parabola

Let the normal at the point P on the parabola

y^{2} = 6 x

pass through the point

(5, - 8)

. If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is:

(A)-3
(B) $- \frac{9}{4}$ ✓ Correct
(C) $- \frac{5}{2}$
(D)-2

Answer: (B)

Solution

For the parabola

y^{2} = 6 x

, we have

4 a = 6

, so

a = \frac{3}{2}

. The equation of the normal at a point

(a t^{2}, 2 a t)

y = - t x + 2 a t + a t^{3}

. Since this passes through

(5, - 8)

, substituting gives

- 8 = - 5 t + 3 t + \frac{3 t ^{3}}{2}

, which simplifies to

t = - 2

. The point P is

(6, - 6)

. The tangent at P has equation

x + 2 y + 6 = 0

. The directrix is

x = - \frac{3}{2}

. Substituting into the tangent equation:

- \frac{3}{2} + 2 y + 6 = 0

, giving

y = - \frac{9}{4}

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Parabola →

More JEE Main questions on Parabola

Let O be the vertex of the parabola $x^{2} = 4 y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C, which is bisected at the point $(1, 2)$ , is:
JEE Main 2026View solution →
Let $y^{2} = 12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $∠ O P A = 9 0^{\circ}$ . Then the locus of the centroid of such triangles OPA is :
JEE Main 2026View solution →
Let one end of a focal chord of the parabola $y^{2} = 16 x$ be (16, 16). If $P (α, β)$ divides this focal chord internally in the ratio $5 : 2$ , then the minimum value of $α + β$ is equal to :
JEE Main 2026View solution →
If the chord joining the points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ on the parabola $y^{2} = 12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2} - y_{1} y_{2}$ is equal t …
JEE Main 2026View solution →
Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2} = 4 x$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio $3 : 1$ , is :
JEE Main 2026View solution →
An equilateral triangle OAB is inscribed in the parabola $y^{2} = 4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is
JEE Main 2026View solution →