Question

If the equations \[\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0\] has equal roots, then

Options

• ab = cd

• ad = bc

• \[ad = \sqrt{bc}\]

• \[ab = \sqrt{cd}\]

Answer 1

The given quadric equation is \[\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0\], and roots are equal.

Here,  `a = (a^2 + b^2), b = -2(ac +bd) and,c = c^2 + d^2`

As we know that `D = b^2 - 4ac`

Putting the value of  `a = (a^2 + b^2), b = -2(ac +bd) and,c = c^2 + d^2`

`={-2 (ac + bd)}^2 - 4 xx (x^2 + b^2) xx (c^2 + d^2)`

`= 4a^2 c^2 + 4b^2 d^2 + 8abcd - 4(a^2 c^2 + a^2d^2 + b^2 c^2 + d^2 d^2)`

`=4a^2 c^2 + 4b^2 d^2 + 8abcd - 4a^2 c^2 - 4a^2 d^2 - 4b^2c^2 - 4b^2d^2`

`= + 8abcd - 4a^2d^2 - 4b^2c^2`

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

The given equation will have equal roots, if D = 0 

` -4(a^2 d^2 + b^2c^2 - 2abcd) = 0`

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

                           `(ad - bc)^2 = 0`

                                 `ad - bc = 0`

                                           `ad = bc`