The equation of circumcircle of triangle formed by lines $7x^2+8xy-y^2=0$ and $2x+y=1$ is $x^2+y^2+2gx+2fy=0$ ,then find $g$ and $f$
I thought if I make equation of circle homogeneous with the help of $2x+y=1$, then resulting curve must be $7x^2+8xy-y^2=0$ but I am not able to get the answer. Is my approach wrong?
The first equation is product of a pair of straight lines intersecting the second line at two points P and Q. We need to find a circle passing through P,Q and point of intersection of the pair of lines.
EDIT1:
The pair of straight lines can be factorized as( because no x,y terms) as
$$ (y - m_1 x)\cdot (y - m_2 x) =0 ,\, m_{1,2}= 4 \pm \sqrt{23} $$
The point of intersection is the origin O!
So required is the circle through $(O,P,Q)$,
where the solved for coordinates of intersection for $P,Q$ are:
$$ (x_1,y_1) = \frac{1} {2+m_1},\frac{ m_1} {2+m_1}, (x_2,y_2) = \frac{1} {2+m_2},\frac{ m_2} {2+m_2}. $$
Unknown coefficients $f,g$ for circle equation are solved using
$$ x_1^2 + y_1^2 + 2 g x_1 + 2 f y_1 = 0,\,x_2^2 + y_2^2 + 2 g x_2 + 2 f y_2 = 0. $$