12. Prove that the equation x2 (a2 + b2)+2x (ac + bd)(2+0 has no real

1.	12. Prove that the equation x2 (a2 + b2)+2x (ac + bd)(2+0 has no real root,if ad # bc.
Answer» It is given: (a^2+ b^2)x^2- 2(ac+bd)x + (c^2+ d^2) = 0 To prove: a / b = c / d PROOF: we know that D = b^2- 4ac b^2= 4ac {-2(ac + bd)}^2= 4{(a^2+ b^2) (c^2+ d^2)} 4(a^2c^2+ b^2d^2+2acbd) = 4(a^2c^2+ a^2d^2+ b^2c^2+ b^2d^2) 2acbd = a^2d^2+ b^2c^2 a^2d^2 + b^2c^2- 2abcd = 0 (ad - bc)^2= 0 ad - bc = 0 ad = bc a / b = c / d hence proved.

12. Prove that the equation x2 (a2 + b2)+2x (ac + bd)(2+0 has no real root,if ad # bc.

Answer»

It is given:

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

To prove:

a / b = c / d

PROOF:

we know that

D = b^2- 4ac

b^2= 4ac

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

4(a^2c^2+ b^2d^2+2acbd) = 4(a^2c^2+ a^2d^2+ b^2c^2+ b^2d^2)

2acbd = a^2d^2+ b^2c^2

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

(ad - bc)^2= 0

ad - bc = 0

ad = bc

a / b = c / d

hence proved.