Question

if the roots of the equation(a²+b²)x²-2(ab+cd)x+(c²+d²)=0 are real and equal,prove that a/b=c/d.please give the solution also

Answer 1

Hey


Your question should be :-


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

Here ,

a = ( a² + b² )

b = -2( ac + bd )

c = ( c² + d² )


Now ,


Given that :- The roots are real and equal .

So ,

D = 0

b² - 4ac = 0

Now , putting value of a , b & c , we get ,


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

=> 4 ( a²c² + b²d² + 2abcd ) - 4 ( a²c² + a²d² + b²c² + b²d² ) = 0


=> 4a²c² + 4b²d² +8abcd - 4a²c² - 4a²d² - 4b²c² - 4b²d² = 0


=> - 4a²d² - 4b²c²+ 8abcd = 0


=> -4 ( a²d² + b²c² - abcd ) = 0


NoTe :- This is in the form of ( a² + b² - 2ab ) , so it will be ( a - b ) ²


=> ( ad - bc ) ² = 0


=> ad - bc = 0


=> ad = bc


=> a / b = c / d


♦ PROVED ♦

thanks :)