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Question

If the roots of the equation a2+b2x2-2ac+bdx+c2+d2=0 are equal, prove that ab=cd.

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Solution

It is given that the roots of the equation a2+b2x2-2ac+bdx+c2+d2=0 are equal.
D=0-2ac+bd2-4a2+b2c2+d2=04a2c2+b2d2+2abcd-4a2c2+a2d2+b2c2+b2d2=04a2c2+b2d2+2abcd-a2c2-a2d2-b2c2-b2d2=0
-a2d2+2abcd-b2c2=0-a2d2-2abcd+b2c2=0ad-bc2=0
ad-bc=0ad=bcab=cd
Hence Proved.

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