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

The combined equation of pair of lines which forms an equilateral triangle with the line 5x+12y−78=0  is  (5x+12y)2−3(12x−5y)2=0The area of the equilateral triangle is c23a2+b2=(78)2352+122=363 square units  If  be the length of the side of the equilateral triangle then the area of the triangle is  34a2It implies that 34a2=363a2=1443a=123=6.9282Therefore, the length of the side is 6.9282