Question

Let a given line L₁ intersect the x and y axes at P and Q respectively. Let another line L₂, perpendicular to L₁, cut the x and y axes at R and S respectively. Show that the locus of the point of intersection of the line PS and QR is a circle passing through the origin.

Answer 1

Let the equation of L₁ be xcosα + y sinα = p₁

Then, any line perpendicular to L₁ is,

Locus of point of intersection of PS and QR can be obtained by eliminating the variable p₂ from Eqs. (i) and (ii).

Which is a circle through origin.