IF THE EQUATION AX^2+BY^2+2FY+2GX+2HXY+C=0 REPRESENTS TWO STRAIGHT LINES, THEN SHOW THAT THE SQUARE OF THE DISTANCE OF THEIR POINT OF INTERSECTION FROM THE ORIGIN IS ((F^2+G^2)-C(A+B))/H^2-AB

Question- IF THE EQUATION AX^2+BY^2+2FY+2GX+2HXY+C=0
REPRESENTS TWO STRAIGHT LINES, THEN SHOW THAT THE SQUARE OF
THE DISTANCE OF THEIR POINT OF INTERSECTION FROM THE ORIGIN IS
((F^2+G^2)-C(A+B))/H^2-AB
Solution- To get the point of intersection, first differentiate the pair equation wrt x, you get,

2AX + 2HY + 2G + (DY / DX) * (2HX + 2BY + 2F) = 0
DY/DX = -(2AX + 2HY + 2G)/(2HX + 2BY + 2F)
Now set
2HX + 2BY + 2F=0
Or HX + BY +F=0 ….(Equation 1)
and
2AX + 2HY + 2G = 0.
AX +HY +G =0 ….(Equation 2)
After solving equation 1 and equation 2 we get
X=


and Y =


So...