Question

Q5. If the equation (1 + m²)x² + 2mcx + (c? - a?) = 0 has equal roots,
prove that c = a’(1 + m?)​

Answer 1

Given Equation -

( 1 + m² ) x² + 2mcx + ( c² - a² ) = 0

Now , this has two equal roots .

Let the roots be â and â respectively .

As the roots Are equal, so Product of Roots :

=> â²

But ,

Product of Roots = ( c² - a² )/ ( 1 + m² )

So,

â² = ( c² - a² ) / ( 1 + m² )

Sum of roots = -b/a

=> 2â = -2mc / ( 1 + m² )

=> â = - mc / ( 1 + m² )

So , we have got the following equations :

=> â = - mc / ( 1 + m² ) .... ¹

=> â² = ( c² - a² ) / ( 1 + m² ) ...... ²

Squaring the first equation ;

[ -mc / ( 1 + m² ) ] ² = ( c² - a² ) / ( 1 + m² )

m²c² / ( 1 + m² )² = ( c² - a ² ) / ( 1 + m² )

=> m²c² = ( 1 + m² )( c² - a² )

=> m²c² = c² - a² + m²c² - m²a²

=> m²a² = c² - a²

=> m²a² + a² = c²

=> c² = a² ( 1 + m² )

=> c = a ( 1 + m² )½ .

Hence Proved

___________________________________________