The equation of the circle passing through the point (1,2) and through the points of inter…

Question

The equation of the circle passing through the point $(1,2)$ and through the points of intersection of $x^2+y^2-4x-6y-21=0$ and $3x+4y+5=0$ is given by:

Accepted Answer

Family through the intersection: $x^2+y^2-4x-6y-21+\lambda(3x+4y+5)=0$. Through $(1,2)$:
$$1+4-4-12-21+\lambda(3+8+5)=0\Rightarrow-32+16\lambda=0\Rightarrow\lambda=2.$$
Circle: $x^2+y^2-4x-6y-21+2(3x+4y+5)=0\Rightarrow x^2+y^2+2x+2y-11=0.$

(Direct check: $1+4+2+4-11=0$ .)

Solutions · 1