Question

Find the envelope of a straight line which moves so that the sum of the squares of the perpendicular drawn to it from two given points is constant.

Answer 1

Take the equation of the line as
$x=\cos \theta +y\sin \theta =p$ ....(1)
and take the axes such that the co ordinates of the fixed points be $(+a,o)$
Then by hypothesis
$p^2+a^2{\cos }^2\theta =$ constant $=c^2$ (say) .....(2)
Putting the value of $p$ from (1), we have
${(x\cos \theta +y\sin \theta )}^2=-a^2{\cos }^2\theta +c^2$
${\tan }^2\theta (y^2-c^2)+2xy\tan \theta +a^2+x^2-c^2=0$
Equating its discriminant equal to zero, the required envelope is the central conic
$x^2{y}^2-(y^2-c^2)(a^2+x^2-c^2)=0$