If p₁ and p₂ denote the lengths of perpendiculars from (2,3) onto the lines given by 15 x^2+31 x y+14 y^2=0 , and if p₁>p₂ , then p₁^2+ 1 74 -p₂^2+ 1 13 is equal to

Step-by-step solution

Given equation, 15 x^2+31 x y+14 y^2=0 aligned 15 x^2+10 x y+21 x y+14 y^2 & =0 5 x(3 x+2 y)+7 y(3 x+2 y) & =0 (3 x+2 y)(5 x+7 y) & =0 aligned Two lines have following equation aligned & 3 x+2 y=0 & 5 x+7 y=0 aligned Perpendicular distance from (2,3) to the line 3 x+2 y=0 is given as | 3(2)+2(3) 9+4 |= 12 13 Perpendicular distance from (2,3) on line 5 x+7 y=0 is given as | 5(2)+7(3) 25+49 |= 31 74 aligned & Given, p₁>p₂ & aligned p₁= 31 74 and p₂ & = 12 13 p₁^2+ 1 74 -p₂^2+ 1 13 & = (31)^2 74 + 1 74 - (12)^2 13 + 1