Question

If the quadratic equation (a2 – b2)x2 + (b2 – c2)x + (c2 – a2) = 0 has equal roots, then which of the following is true? (A) b2 + c2 = a2 (B) b2 + c2 = 2a2 (C) b2 – c2 = 2a2 (D) a2 = b2 + 2c2

Answer 1

$( {a}^{2} - {b}^{2} ) {x}^{2} + ( {b}^{2} - {c}^{2} )x + ( {c}^{2} - {a}^{2} ) = 0 \\ \\ the \: sum \: of \: coefficients \: and \: constant \\ term \: gives \\ ( {a}^{2} - {b}^{2} ) + ( {b}^{2} - {c}^{2} ) + ( {c}^{2} - {a}^{2} ) = 0 \\ \\ so \: \: x = 1 \: is \: one \: of \: the \: root. \\ since \: roots \: are \: equal \: \: other \: root \: is \\ also \: \: 1 \\ \\ thus \: \: product \: \: of \: \: roots = 1 \\ \frac{ {c}^{2} - {a}^{2} }{ {a }^{2} - {b}^{2} } = 1 \\ {c}^{2} - {a}^{2} = {a }^{2} - {b}^{2} \\ {b}^{2} + {c}^{2} = 2 {a}^{2} \\ option \: \: (B)$