if the roots of the equation (a×a+b×b)x×x-2(ac+bd)x+(c×c+ d×d)=0 are equal ,prove that a/b=c/d.
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as we know the if the roots of the quadratic equation js zero then discriminant of the equation is zero
thus d^2-4ac= 4(AC+bd)^2-4(a^2+b^2)(c^2+d^2)=0
4(a^2c^2+b^2d^2+2abcd-a^2c^2-a^2d^2-b^2c^2-b^2d^2)=0
=4(2abcd-a^2d^2-b^2c^2)=0
-(ad-bc)^2=0
ad-bc=0
ad=bc
a/b=c/d
thus d^2-4ac= 4(AC+bd)^2-4(a^2+b^2)(c^2+d^2)=0
4(a^2c^2+b^2d^2+2abcd-a^2c^2-a^2d^2-b^2c^2-b^2d^2)=0
=4(2abcd-a^2d^2-b^2c^2)=0
-(ad-bc)^2=0
ad-bc=0
ad=bc
a/b=c/d
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