Question

A point moves so that the sum of the squares of its distances from two intersecting straight lines is constant. Prove that its locus is an ellipse.

Answer 1

OA and DB
`y=ntantheta,y=-xtantheta`
`xsintheta-ycostheta=0`
`xsintheta+ycostheta=0`
`PL^2+PM^2=lambda^2`(const)
`((hsintheta-kcostheta)/sqrt(sin^2theta+cos^2theta))^2+((hsintheta+kcostheta)/sqrt(sin^2theta+cos^2theta))^2=lambda^2`
`h^2sin^2theta+k^2cos^2theta=lambda^2`
`h^2/((lambda^2)/(sin^2theta))+k^2/((lambda^2/cos^2theta))=1`
`(x^2)/(lambda/sintheta)^2+k^2/(lambda/costheta)^2=1`.