A chord PQ is a normal to the parabola y2 = 4ax at P and subtends a right angle at the vertex. If SQ = λSP where S is the focus then the value of λ, is

Engineering

Mathematics

Chord of Parabola

Question

A chord PQ is a normal to the parabola y² = 4ax at P and subtends a right angle at the vertex. If SQ = λSP where S is the focus then the value of λ, is

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We have, $\frac{2}{t_{1}} \cdot \frac{2}{t_{2}}$ = –1 ⇒ t₁ t₂ = –4 ; also t₂ = – t₁ – $\frac{2}{t_{1}}$    ....(1)
t₁t₂ = – $t_{1}^{2}$ – 2   ⇒ – 4 + 2 = – $t_{1}^{2}$ ⇒ $t_{1}^{2}$ = 2
Also, $t_{2}^{2} = t_{1}^{2} + \frac{4}{t_{1}^{2}} + 4$    squaring (1)
⇒   $t_{2}^{2}$ = 2 + 2 + 4 = 8
Now,   SQ = a(1 + $t_{2}^{2}$ ) = a(1 + 8) = 9a and SP = a(1 + $t_{1}^{2}$ ) = a(1 + 2) = 3a
∴   $\frac{SQ}{SP}$ = 3 ⇒ SQ = 3SP ⇒ λ = 3 Ans.

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