Question

If the equation:-
$\sf (a² + b²)x - 2 (ac + bd)x + c² + d² = 0$
has equal roots, then:-
$(a) ab = cd$
$(b) ad = bc$
$(c)ad = \sqrt{bc}$
$(d) ab = \sqrt{cd}$
Explain how OPTION (d) is CORRECT.

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Answer 1

CORRECT QUESTION.

If the equation

$( {a}^{2} + {b}^{2}) {x}^{2} - 2(ac + bd)x + {c}^{2} + {d}^{2} = 0$

has equal roots then,

Explanation.

If the roots is equal

Therefore,

D = 0

$\bold{b}^{2} - 4ac = 0$

Equation are,

$( {a}^{2} + {b}^{2}) {x}^{2} - 2(ac + bd)x + {c}^{2} + {d}^{2} = 0$

$\bold( - 2(ac + bd) {}^{2}) - 4( {a}^{2} + {b}^{2})( {c}^{2} + {d}^{2} ) = 0$

$\bold4( {a}^{2} {c}^{2} + {b}^{2} {d}^{2} + 2abcd - {a}^{2} {c}^{2} - {a}^{2} {d}^{2} - {b}^{2} {c}^{2} - {b}^{2} {d}^{2}) = 0$

$2abcd - {a}^{2} {d}^{2} - {b}^{2} {c}^{2} = 0$

$\bold{a}^{2} {d}^{2} + {b}^{2} {c}^{2} - 2abcd = 0$

$\bold(ad - bc) {}^{2} = 0$

$\bold{ad \: = bc}$

Hence,

Option [B] is correct

Note = also see the image attachment.