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Equation of a Chord When Its Mid Point Is Given

Let the equat...

Question

Let the equations of tangents to parabola $y^{2} = 4 a x$ which are drawn from the point $(2 a, 3 a)$ be $k x + m y + 4 a = 0; p x - h y + a = 0$ .Find $| k + m + p + h |$ ?

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Solution

Any tangent to parabola is given by, $y = m x + \frac{a}{m}$
Given it passes through $(2 a, 3 a) \Rightarrow 3 a = 2 a m + \frac{a}{m}$
$\Rightarrow 2 m^{2} - 3 m + 1 = 0 \Rightarrow m = 1, \frac{1}{2}$
Hence tangents are $x - y + a = 0, x - 2 y + 4 a = 0$

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