The product of the perpendicular distances from any point on the hyperbola x^2 a^2 - y^2 b^2 =1 to its asymptotes is

Step-by-step solution

Let (a , b ) be any point on the hyperbola x^2 a^2 - y^2 b^2 =1 The equation of the asymptotes of the given hyperbola are ( x a + y b )=0 and ( x a - y b )=0 Now, P₁= length of the perpendicular from (a , b ) on ( x a + y b )=0 P₁= + 1 a^2 + 1 b^2 ...(i) P₂= length of the perpendicular from (a , b ) on ( x a - y b )=0 P₂= - 1 a^2 + 1 b^2 ...(ii) P₁ P₂= ( + ) 1 a^2 + 1 b^2 ( - ) 1 a^2 + 1 b^2 = ( ^2 - ^2 ) ( 1 a^2 + 1 b^2 ) = 1 (a^2+b^2 ) a^2 b^2 P₁ P₂= a^2 b^2 a^2+b^2