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प्रश्न
If the roots of the equation (a2 + b2) x2 – 2(ac + bd) x + (c2 + d2) = 0 are equal, prove that `a/b = c/d`
उत्तर
The given quadratic equation is (a2 + b2)x2 − 2(ac + bd)x + (c2 + d2) = 0.
If the roots of given quadratic equation are equal, then its discriminant is zero.
`:. D = [-2(ac + bd)^2] - 4(a^2 + b^2)(c^2 + d^2) = 0`
`=> 4(a^2c^2+ b^2d^2 + 2abcd) - 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0`
`=> 4(a^2c^2 + b^2d^2 + 2abcd - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2) = 0`
`=>2abcd - a^2d^2 - b^2c^2 = 0`
`=> a^2d^2 + b^2c^2 - 2abcd = 0`
`=>(ad - bc)^2 = 0`
⇒ ad - bc = 0
⇒ad = bc
`=> a/b = c/d`
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