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`

Answer 1

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.