Question

Extremities of a diagonal of a rectangle are $(0,0)$ and $(4,3)$. Find the equations of the tangents to the circumcircle of the rectangle which are parallel to this diagonal.

Answer 1

Circumcircle of rectangle will be a circle on $AC$ as diameter $x(x-4)+y(y-3)=0$
or $x^2+y^2-4x-3y=0$
Centre $(2,3/2)$ and radius $5/2$
Equation of $AC$ is $3x-4y=0$
Any line parallel to it is $3x-4y+\lambda =0$
If it is a tangent, then $p=r$ gives
$\frac{6-6+\lambda }{5}=\pm \frac{5}{2}\therefore \lambda =\pm \frac{25}{2}$
$\therefore 3x-4y\pm \frac{25}{2}=0$ are the required tangents.