Question

Write the condition, if the roots of ax2 + bx+c = 0 are sine, cose​

Answer 1

Answer:

a² + 2ac = 4b²

Step-by-step explanation:

Given---> Roots of ax² + bx + c = 0 are Sinθ and Cosθ.

To find---> Required condition

Solution---> ax² + bx + c = 0

Roots of equation are Sinθ and Cosθ,

We know that

Sum of roots = - ( 2b / a )

=> Sinθ + Cosθ = - ( 2b / a ) ...........(1)

Product of roots = c / a

Sinθ Cosθ = c / a ..................(2 )

By equation ( 1 ) we get

Sinθ + Cosθ = - ( 2b / a )

Squaring both sides we get

=> ( Sinθ + Cosθ )² = ( - 2b / a )²

We have an identity as follows

( a + b )² = a² + b² + 2ab , using it ,we get

=> Sin²θ + Cos²θ + 2 Sinθ Cosθ = 4b²/ a²

We have an identity as follows

Sin²θ + Cos²θ = 1 , we get

=> 1 + 2 ( Sinθ Cosθ ) = 4b² / a²

Putting Sinθ Cosθ = c / a

=> 1 + 2 ( c / a ) = 4b² / a²

Multiplying whole relation by a² , we get

=> a² ( 1) + 2a² ( c / a ) = a² ( 4b² / a² )

=> a² + 2ac = 4b²

It is the required condition