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If the Roots of the Equation (A2 + B2)X2 − 2 (Ac + Bd)X + (C2 + D2) = 0 Are Equal, Prove that `A/B=C/D`. - CBSE Class 10 - Mathematics

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Question

If the roots of the equation (a2 + b2)x2 − 2 (ac + bd)x + (c2 + d2) = 0 are equal, prove that `a/b=c/d`.

Solution

The given quadric equation is (a2 + b2)x2 − 2 (ac + bd)x + (c2 + d2) = 0, and roots are real

Then prove that `a/b=c/d`.

Here,

a = (a2 + b2), b = -2 (ac + bd) and c = (c2 + d2)

As we know that D = b2 - 4ac

Putting the value of a = (a2 + b2), b = -2 (ac + bd) and c = (c2 + d2)

D = b2 - 4ac

= {-2(ac + bd)}2 - 4 x (a2 + b2) x (c2 + d2)

= 4(a2c2 + 2abcd + b2 + d2) - 4(a2c2 + a2d2 + b2c2 + b2d2)

= 4a2c2 + 8abcd + 4b2d2 - 4a2c2 - 4a2d2 - 4b2c2 - 4b2d2

= -4a2d2 - 4b2c2 + 8abcd

= -4(a2d2 + b2c2 - 2abcd)

The given equation will have real roots, if D = 0

-4(a2d2 + b2c2 - 2abcd) = 0

a2d2 + b2c2 - 2abcd = 0

(ad)2 + (bc)2 - 2(ad)(bc) = 0

(ad - bc)2 = 0

Square root both sides we get,

ad - bc = 0

ad = bc

`a/b=c/d`

Hence `a/b=c/d`

  Is there an error in this question or solution?

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Solution If the Roots of the Equation (A2 + B2)X2 − 2 (Ac + Bd)X + (C2 + D2) = 0 Are Equal, Prove that `A/B=C/D`. Concept: Nature of Roots.
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