If l x+m y=1 is a normal to the hyperbola x^2 a^2 - y^2 b^2 =1 , then a^2 m^2-b^2 l^2=

Step-by-step solution

Given equation of normal, l x+m y=1 y= -l m + 1 m Which is of the form y=m x+c Since, the lines y=m x+c will be normal to the hyperbola aligned & x^2 a^2 - y^2 b^2 =1, if c^2= m^2 (a^2+b^2 )^2 a^2-b^2 m^2 & ( 1 m )^2= (- l m )^2 (a^2+b^2 )^2 a^2-b^2 (- l m )^2 aligned aligned & 1 m^2 = l^2 m^2 (a^2+b^2 )^2 m^2 a^2-l^2 b^2 m^2 & 1 m^2 = l^2 (a^2+b^2 )^2 m^2 a^2-l^2 b^2 & a^2 m^2-b^2 l^2=l^2 m^2 (a^2+b^2 )^2 aligned