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Let alpha, beta are the root of equation...

Let `alpha, beta` are the root of equation `acostheta + bsintheta = c`.
Which of the following is/are true.

A

`sinalpha + sinbeta= (2bc)/(a^(2) + b^(2))`

B

`sinalpha + sinbeta= (2bc)/(b^(2) + c^(2))`

C

`tan'(alpha)/(2) + tan'(beta)/(2) = (b)/(a + c)`

D

`tan'(alpha)/(2) + tan"(beta)/(2) = (2b)/(a + c)`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`"Sum" = sinalpha + sinbeta = (2bc)/(a^(2) + b^(2))`
Product `= sinalpha sinbeta = (c^(2) - a^(2))/(c^(2) - a^(2))`
`tan ((alpha)/(2)) + tan ((beta)/(2)) = (2b)/(a + c)`
bacause equation is `(a^(2) - b^(2))sin^(2)theta - 2bc sintheta + c^(2) - a^(2) = 0`
or `(a + c)tan^(2)((theta)/(2)) - 2btan((theta)/(2)) + c - a = 0`
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Knowledge Check

  • Let alpha, beta are the root of equation acostheta + bsintheta = c . If alpha = 30^(@) and beta = 60^(@) such that a, b, c represent sides of a DeltaABC then

    A
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    B
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    C
    `ABC` is right angle triangle
    D
    `ABC` is obtuse angle triangle

  • Let alpha, and beta are the roots of the equation x^(2)+x +1 =0 then

    A
    `alpha^(2) +beta^(2) =4`
    B
    `(alpha - beta)^(2)=3`
    C
    `alpha^(3) +beta^(3)=2`
    D
    `alpha^(4) +beta^(4) = 1`

  • If x=asintheta-bcostheta,y=acostheta+bsintheta , then which of the following is true ?

    A
    `(x^(2))/(y^(2))+(a^(2))/(b^(2))=1`
    B
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    C
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    D
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