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Question
If the roots of the equation `(a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0`are equal, prove that `a/b=c/d`
Solution
It is given that the roots of the equation `(a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0 are equal
∴`D=0`
⇒`[-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`
⇒`(-a^2d^2+2abcd-b^2c^2)=0`
⇒`-(a^2d^2-2abcd+b^2c^2)=0`
⇒`(ad-bc)^2=0`
⇒`ad-bc=0`
⇒`ad=bc`
⇒`a/b=c/d`
Hence proved.
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