If O,A,B  are vertices of a triangle whose sides are given by  5x+12y2−312x−5y2=0 and 5x+12y−78=0   then AB  =

The triangle formed by the lines 5x+12y2−312x−5y2=0  and  5x+12y−78=0  is equilateral triangle. The area of The area of the equilateral triangle formed by the lines ax+by+c=0  and the pair of lines ax+by2−3bx−ay2=0  is  c23(a2+b2) Hence, the area of the triangle is  7823(52+122)If the length of the side of the equilateral triangle is a  then its area is  3a24it implies that3a24=782352+122a2=4⋅78⋅783169=4⋅6⋅63=48Therefore,a=43