The straight lines joining origin and the point of intersections of the curve $y^{2}$ =4x with the line x+y+n = 0 are perpendicular. Find the value of $n^{2}$ .

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Solution

The combined equation of the straight lines can be obtained by the method of homogenization.

$\Rightarrow y^{2} = 4 x \frac{x + y}{- n}$

$\Rightarrow - n y^{2} = 4 x^{2} + 4 x y$

$\Rightarrow 4 x^{2} + 4 x y + n y^{2} = 0$

These straight lines are perpendicular if

4 + n = 0

$\Rightarrow n = - 4$

$\Rightarrow n^{2} = 16$

The tan of the angle between the pair of straight lines $a x^{2} + 2 b x y + b y^{2} = 0$ is given by

$tan θ = ∣ ∣ ∣ \frac{2 \sqrt{b^{2} - a b}}{a + b} ∣ ∣ ∣$

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