The product of the lengths of the perpendiculars drawn from the point (-1,5) to the pair of lines 2 x^2-x y-3 y^2+6 x+y+4=0 is

Step-by-step solution

The pair of straight line array rl 2 x^2+(-y+6) x+ (-3 y^2+y+4 )=0 & 2 x^2-x y-3 y^2+6 x+y+4=0 x & = (y-6) (y-6)^6-4 2 (-3 y^2+y+4 ) 2 2 & = (y-6) y^2+36-12 y+24 y^2-8 y-32 4 & = (y-6) 25 y^2+4-20 y 4 & = (y-6) (5 y-2)^2 4 x & = (y-6) (5 y-2) 4 So, x & = y-6+(5 y-2) 4 and x= y-6-5 y+2 4 4 x & =6 y-8 and 4 x=-4 y-4 2 x & -3 y+4=0 and x+y+1=0 array Now, product length of perpendicular from (-1,5) to both lines are aligned & P₁ P₂= | -2-3(5)+4 2^2+(-3)^2 | | -1+5+1 1^2+1^2 | & = 13 13 5 2 = 65 26 aligned