Question

Prove that the locus of the point of intersection of the lines
$xcos\alpha +ysin\alpha =a$
and $xsin\alpha -ycos\alpha =b$
is a circle whatever $\alpha$ may be.

Answer 1

In order to find the locus of the point of intersection we -have to eliminate the variable a between the lines for which we square and add.
$\therefore$ x${}^2$ (sin${}^2$ $\alpha$ + cos${}^2$ $\alpha$)+y${}^2$ (sin${}^2$ $\alpha$ + cos${}^2$.$\alpha$) = a${}^2$ + b${}^2$
or x${}^2$ + y${}^2$ = a${}^2$ + b${}^2$ which represents a circle.