If root of the equations (a2+b2)x²-2(ac+bd)x . . .

Question : If root of the equations (a²+b²)x²-2(ac+bd)x+c²+d²=0 are equal then prove that a/b=c/d.

Doubt by Vanshika

OR 
Show that if the roots of the following quadratic equation in x (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal then ad=bc.

Doubt by Zoha

Solution : 

(a²+b²)x²-2(ac+bd)x+c²+d²=0

Here 
A=a²+b²

B=-2(ac+bd)

C=c²+d²

D=B²-4AC
D=0 (Roots are equal)
0=B²-4AC
0=[-2(ac+bd)]²-4(a²+b²)(c²+d²)
0=4(ac+bd)²-4(a²+b²)(c²+d²)
0=4[(ac+bd)²-(a²+b²)(c²+d²)]
0/4=(ac+bd)²-(a²+b²)(c²+d²)
0=a²c²+b²d²+2(ac)(bd)-[a²(c²+d²)+b²(c²+d²)]
0=a²c²+b²d²+2(ac)(bd)-[a²c²+a²d²+b²c²+b²d²]
0=a²c²+b²d²+2(ac)(bd)-a²c²-a²d²-b²c²-b²d²

0=2(ac)(bd)-a²d²-b²c²
0=-a²d²-b²c²+2(ac)(bd)
0=-[a²d²+b²c²-2(ac)(bd)]
0=[ad-bc]²

[∵x²+y²-2xy=(x-y)²]
±√0=ad-bc
0=ad-bc
-ad=-bc
ad=bc
a/b=c/d

Hence proved.