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If Sin α and Cos α Are the Roots of the Equations Ax2 + Bx + C = 0, Then B2 = - CBSE Class 10 - Mathematics

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Question

If sin α and cos α are the roots of the equations ax2 + bx + c = 0, then b2 =

  •  a2 − 2ac

  •  a2 + 2ac

  • a2 − ac

  • a2 + ac

Solution

The given quadric equation is `ax^2 + bx + c = 0`, and `sin alpha and cos beta` are roots of given equation.

And, a = a,b = b and, c = c

Then, as we know that sum of the roots

 `sin alpha + cos beta - (-b)/a`…. (1)

And the product of the roots

`sin alpha .cos beta =c/a`…. (2)

Squaring both sides of equation (1) we get

`(sin alpha + cos beta)^2 = ((-b)/a)^2`

`sin^2 alpha + cos^2 beta + 2 sin alpha cos beta = b^2/a^2`

Putting the value of `sin^2 alpha + cos^2 beta = 1`, we get

`1 + 2 sin alpha cos beta = b^2/a^2`

`a^2 (1+2 sin alpha cos beta) = b^2`

Putting the value of`sin alpha.cos beta = c/a` , we get

`a^2 (1 + 2 c/a) = b^2`

`a^2 ((a+2c)/a) = b^2`

        `a^2 + 2ac =b^2`

Therefore, the value of `b^2 = a^2 + 2ac`.

  Is there an error in this question or solution?

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Solution If Sin α and Cos α Are the Roots of the Equations Ax2 + Bx + C = 0, Then B2 = Concept: Solutions of Quadratic Equations by Factorization.
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