Question

If the equation (1+ m²)x² +2mcx +(c²-a²) =0 has equal roots prove that c² = a² ( 1 + m²)


(CBSE Class -10th SP)

Answer 1

HELLO DEAR,


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

=> b2 - 4ac = 0

=> (2 mc)2 - 4(1 + m2)(c2 - a2) = 0

=> 4m2c2 - 4(c2 - a2 + m2c2 - m2a2) = 0

=> 4m2c2 - 4c2 + 4a2 - 4m2c2 + 4m2a2 = 0


I HOPE ITS HELP YOU DEAR,
THANKS

=>  4m2a2 - 4c2 + 4a2 = 0

=> m2a2 - c2 + a2 = 0
=> a2(1 + m2) - c2 = 0 

=> c2  = a2(1 + m2)

Answer 2

Hey there,

Conditions implemented → For equal roots

Hence, D=0 , b² = 4ac

proceeding in that manner,

Given equation,

(1 + m²)x² + 2mcx + ( c² - a² ) =0

For equal roots,

4m²c² = 4( 1 + m² ) ( c² - a² )

4m²c² = 4c² - 4a² + 4c²m² - 4a²m²

4c² - 4a²m² - 4a² = 0

 4 [ c² - a²m² - a² ] = 0

 c² = a²m² + a²

 c² = a² [ 1 + m² ] 

Hope this helps!