Question

The extremities of a diameter of a circle are $(1,2)$ and $(3,4)$. Find its equation. Also determine the equations of tangents which are parallel to this diameter.

Answer 1

The slope of given line is $1$ and circle on this line as diameter is
$x^2+y^2-4x-6y+11=0$ or $(2,3),\sqrt{2}$.
Any line parallel to given diameter is $y=x+c$ or $x-y+c=0$. The condition of tangency $p=r$ gives $c=3,-1$.
$\therefore x-y+3=0$ and $x-y-1=0$ are tangents.