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# If the Equations ( a 2 + B 2 ) X 2 − 2 ( a C + B D ) X + C 2 + D 2 = 0 Has Equal Roots, Then - CBSE Class 10 - Mathematics

ConceptSolutions of Quadratic Equations by Factorization

#### Question

If the equations $\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0$ has equal roots, then

• ab = cd

• $ad = \sqrt{bc}$

• $ab = \sqrt{cd}$

#### Solution

The given quadric equation is $\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0$, and roots are equal.

Here,  a = (a^2 + b^2), b = -2(ac +bd) and,c = c^2 + d^2

As we know that D = b^2 - 4ac

Putting the value of  a = (a^2 + b^2), b = -2(ac +bd) and,c = c^2 + d^2

={-2 (ac + bd)}^2 - 4 xx (x^2 + b^2) xx (c^2 + d^2)

= 4a^2 c^2 + 4b^2 d^2 + 8abcd - 4(a^2 c^2 + a^2d^2 + b^2 c^2 + d^2 d^2)

=4a^2 c^2 + 4b^2 d^2 + 8abcd - 4a^2 c^2 - 4a^2 d^2 - 4b^2c^2 - 4b^2d^2

= + 8abcd - 4a^2d^2 - 4b^2c^2

 = -4(a^2 d^2 + b^2c^2 - 2abcd)

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

 -4(a^2 d^2 + b^2c^2 - 2abcd) = 0

 a^2 d^2 +b^2 c^2 - 2abcd = 0

(ad - bc)^2 = 0

ad - bc = 0

ad = bc

Is there an error in this question or solution?

#### APPEARS IN

Solution If the Equations ( a 2 + B 2 ) X 2 − 2 ( a C + B D ) X + C 2 + D 2 = 0 Has Equal Roots, Then Concept: Solutions of Quadratic Equations by Factorization.
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