Question

If the roots of the equation (a² + b²) x² – 2(ac + bd) x + (c² + d²) = 0 are equal, prove that `a/b = c/d`

Answer 1

The given quadratic equation is (a² + b²)x² − 2(ac + bd)x + (c² + d²) = 0.

If the roots of given quadratic equation are equal, then its discriminant is zero.

`:. D = [-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`

`=>2abcd - a^2d^2 - b^2c^2 = 0`

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

`=>(ad - bc)^2 = 0`

⇒ ad - bc = 0

⇒ad = bc

`=> a/b = c/d`