Question

if the roots of an eq. (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

Answer:

a/b = c/d

Hence, it is proved.

Step-by-step explanation:

(a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0

This equation is in form of a²x + bx + c = 0.

.°. The roots are equal.

So, D = b² - 4ac = 0

Here,

a = (a² + b²)

b = 2(ac + bd)

c = (c² + d²)

=> b² - 4ac = 0

=> [{2(ac + bd)}² - 4(a² + b²)(c² + d²)] = 0

=> (a²b² + c²d² + 2acbd) - (a²c² - a²d² - b²d² - b²c²) = 0

=> - a²c² - b²d² + 2acbd = 0

=> - (a²c² + b²d² + 2acbd) = 0

Negative sign goes to the Right Hand Side.

=> a²c² + b²d² - 2acbd = 0

By using the identity of (a - b)² = a² + b² - 2ab.

=> (ad - bc)² = 0

=> (ad - bc) = √0

=> ad - bc = 0

=> ad = bc

So, a/b = c/d

Hence, it is proved.