If (m+3 n)(3 m+n)=4 h^2 , then the acute angle between the lines represented by m x^2+2 h x y+n y^2=0 is

Step-by-step solution

Given lines are m x^2+2 h x+n y^2=0 Let m ₁ and m ₂ be the slopes of the lines aligned & m ₁+ m ₂= -2 ~h n and m ₁ ~m ₂= m n & Now ( m ₁- m ₂ )^2= ( m ₁+ m ₂ )^2-4 ~m ₁ ~m ₂ & = ( -2 ~h n )^2- 4 ~m n = 4 ~h ^2-4 mn n ^2 = 3 ~m ^2+6 mn +3 n ^2 n ^2 [ From (1)] & = 3( ~m + n )^2 n ^2 & m ₁- m ₂= 3 ( ~m + n ) n aligned Let be the required angle. aligned & Now = | m ₁- m ₂ | 1+ m ₁ ~m ₂ & = ( 3 ( ~m + n ) n ) (1+ m n ) = 3 ( ~m + n ) n n m + n = 3 =60^ = ^ c 3 aligned