Question

Tangent is drawn at any point (p, q) on the parabola y² = 4ax. Tangents are drawn from any point on this tangent to the circle x² + y² = a², such that the chords of contact pass through a fixed point (r, s) then p, q, r s can hold the relation –

• A. r²q = 4p²s
• B. rq² = 4ps²
• C. rq² = –4ps²
• D. r²q = –4p²s

Answer 1

Answer: (C)

rq² = –4ps²

Answer 2

Equation of tangent yq = 2a (x + p)

Let (x₁, y₁) on this tangent then

y₁q = 2a (x₁ + p)

chord of contact for circle

xx₁ + yy₁ = a²

passing through (r, s)

rx₁ + sy₁ = a²

also xx₁ + y $\frac{2a}{q}$ (x₁ + p) – a² = 0

x₁ $\left( x + \frac{2ay}{q} \right)$+ $\left( \frac{2ap}{q}y - a^{2} \right)$= 0

x₁ (qx + 2ay) + (2apy – a²q) = 0

qx + 2ay = 0

2apy = a² q

̃ qpx + a² q = 0 ̃ x = –$\frac{a^{2}}{p}$, y =$\frac{aq}{2p}$

so r = –$\frac{a^{2}}{p}$ and s =$\frac{aq}{2p}$ ̃ 4ps² = – rq²