Paragraph: The roots of the quadratic equation a^2 (b^2-c^2 ) x^2+b^2 (c^2-a^2 ) x+c^2 (a^2-b^2 )=0 are equal (a^2 b^2 c^2 ) . Question: Which one of the following statements is correct?

Step-by-step solution

The given equation is as follows, a^2 (b^2-c^2 ) x^2+b^2 (c^2-a^2 ) x+c^2 (a^2-b^2 )=0 ....(i) Since, equation has equal roots D=0 aligned & [b^2 (c^2-a^2 ) ]^2-4 a^2 c^2 (b^2-c^2 ) (a^2+b^2 )=0 & b^4 (c^2-a^2 )^2-4 a^2 c^2 (b^2-c^2 ) (a^2-b^2 )=0 aligned Solving above equation, we have, aligned & b^4 (c^2+a^2 )^2=4 a^4 c^4 b^2 (a^2+c^2 )^2=2 a^2 c^2 & 2 b^2 = 1 a^2 + 1 c^2 aligned a^2, b^2, c^2 and in H.P. Option (c) is the correct option.