Question

If the quadratic equation `(1+m^2)x^2+2mcx+(c^2-a^2)=0` has equal roots, prove that  `c^2=a^2(1+m^2)`  

Answer 1

Given: 

`(1+m^2)x^2+2mcx+(c^2-a^2)=0` 

Here, 

`a=(1+m^2), b=2mc and c=(c^2-a^2)` 

It is given that the roots of the equation are equal; therefore, we have: 

`D=0` 

⇒` (b^2-4ac)=0` 

⇒ `(2m)^2-4xx(1+m^2)xx(c^2-a^2)=0` 

⇒`4m^2c^2-4(c^2-a^2+m^2c^2-m^2a^2)=0` 

⇒` 4m^2c^2-4c^2+4a^2-4m^2c^2+4m^2a^2=0` 

⇒`-4c^2+4a+4m^2a^2=0` 

⇒`a^2+m^2a^2=c^2` 

⇒`a^2(1+m^2)=c^2` 

⇒`c^2=a^2(1+m^2)` 

Hence proved