Question

If the roots of the equation ( a^2 + b^2 ) x^2 - 2 ( ac+ bd )x + ( c^2 + d^2 ) = 0 have equal roots . prove that a/b = c/d .

Answer 1

Given Equation is (a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0

On comparing with ax^2 + bx + c = 0, we get a = a^2 + b^2, b = -2(ac + bd) , c = c^2 + d^2.

Given that The equation has equal roots.

= > D = 0

= > b^2 - 4ac = 0

= > b^2 = 4ac.

= > (-2(ac + bd))^2 = 4(a^2 + b^2)(c^2 + d^2)

= > 4(a^2c^2 + b^2d^2 + 2abcd) = 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2)

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

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

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

= > (ad - bc)^2 = 0

= > ad = bc

= > a/b = c/d

Hope this helps!