Question

If two tangents to the parabola $y^{2} = 4 a x$ from a point $P$ make angles $θ_{1}$ and $θ_{2}$ with the axis of the parabola, then find the locus of $P$ in each of the following cases.

(iii) $\tan θ_{1} + \tan θ_{2} = λ$ (is constant)

Accepted Answer

$y^{2} = 4 a x . . . (i)$

Point $P : (h, k)$

Tangent to $y^{2} = 4 a x$ is $T : y = m x + \frac{a}{m} . . . (ii)$

It passes through $(h, k) \Rightarrow m^{2} h - m k + a = 0 . . . (iii)$

Let the roots to $(iii)$ be $m_{1}, m_{2}$

$m_{1} + m_{2} = \frac{k}{h}; m_{1} m_{2} = \frac{a}{h}$

$\Rightarrow m_{1} - m_{2} = \frac{1}{h} \sqrt{k^{2} - 4 a h} . . . (v)$

Locus of point of intersection

$\Rightarrow \tan β = \pm \frac{m_{1} + m_{2}}{1 - m_{1} m_{2}} = \frac{k}{h - a} . . . (iv)$

Given tangent makes angle $θ_{1} & θ_{2}$ with axis

$m_{1} = \tan θ_{1} & m_{2} = \tan θ_{2}$

Now, $\tan β = \pm \frac{\tan θ_{1} + \tan θ_{2}}{1 - \tan θ_{1} + \tan θ_{2}} = \pm \tan (θ_{1} + θ_{2}) . . . (vii)$

Putting values from $(vii) to (vi)$

$\Rightarrow y^{2} = \tan^{2} (θ_{1} + θ_{2}) {(x - a)}^{2} . . . (viii)$

$∵$ $\tan θ_{1} + \tan θ_{2} = λ$

Equation $(vii) \Rightarrow y^{2} = {[\frac{λ}{1 - \tan θ_{1} \tan θ_{2}}]}^{2} {(x - a)}^{2}$

$\Rightarrow y^{2} {(1 - \tan θ_{1} \tan θ_{2})}^{2} = λ^{2} {(x - a)}^{2}$

$[as \tan θ_{1} \tan θ_{2} = m_{1} m_{2} = \frac{a}{h}]$

$\Rightarrow y^{2} {(1 - \frac{a}{x})}^{2} = λ^{2} {(x - a)}^{2}$

$∴ y^{2} = λ^{2} x^{2}$

Exercise-1

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