Question

Find the envelope of a straight line which moves so that the difference of these squares is constant.

Answer 1

The difference of squares of the perpendiculars let drop from $(\pm a,0)$ is
${(a\cos \theta -p)}^2-{(a\cos \theta +p)}^2=$ constant (by hypothesis)
$\Rightarrow 4p\left(a\cos \theta \right)=$ constant
$\Rightarrow p\cos \theta =\lambda$ (constant) .....(1)
(as $\theta$ is already constant)
Now multiplying the equation of the line
$x\cos \theta +y\sin \theta =p$ by $\cos \theta$, we get
$x{\cos }^2\theta +y\sin \theta \cos \theta =p\cos \theta =\lambda ({\cos }^2\theta +{\sin }^2\theta )$ by (1)
$\lambda {\tan }^2\theta -y\tan \theta +(\lambda -x)=0$

Now equating its determinant equal to zero the required envelope is the curve $y^2=4\lambda (\lambda -x)$ which is a parabola.