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Q.

The locus of the point of intersection of two tangents to the parabola $y^{2} = 4 a x$ which with the tangent at the vertex form a triangle of constant area $c^{2},$ is the curve $x^{2} (y^{2} - 4 a x) = λ c^{4}$ .
Then $λ$ =

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answer is 4.

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Detailed Solution

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$p = (a t_{1} t_{2}, a (t_{1} + t_{2})) = (h, k) [Q = (0, a t_{1}), R = (0, a t_{2})] A r e a o f Δ P Q R = \frac{1}{2} | a (t_{2} - t_{1}) a t_{1} t_{2} | \Rightarrow c^{2} = \frac{a^{4}}{4} t_{1}^{2} t_{2}^{2} ({(t_{1} + t_{2})}^{2} - 4 t_{1} t_{2})$
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$= \frac{a^{4}}{4} . \frac{h^{2}}{a^{2}} ({(\frac{k}{a})}^{2} - \frac{4 h}{a}) \Rightarrow 4 c^{4} = h^{2} (k^{2} - 4 a h) x^{2} (y^{2} - 4 a x) = 4 c^{4}$

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The locus of the point of intersection of two tangents to the parabola y2=4ax which with the tangent at the vertex form a triangle of constant area c2, is the curve x2(y2−4ax)=λc4 .Then λ=