IF THE EQUATION AX^2+BY^2+2FY+2GX+2HXY+C=0 REPRESENTS TWO STRAIGHT LINES, THEN SHOW THAT THE SQUARE...

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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

David · Accepted Answer

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  ….

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

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