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Relation between Roots and Coefficients for Quadratic

If tan α, tan...

Question

If $t a n α, t a n β$ are the roots of the equation $x^{2}$ + ax + b = 0 then the value of $s i n^{2} (α + β) + a s i n (α + β) c o s (α + β) + b c o s^{2} (α + β) =$

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$\frac{1}{a} + b$

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Solution

The correct option is B
b

$t a n α + t a n β = - a, t a n α t a n β = b t a n (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β} t a n (α + β) = \frac{- a}{1 - b} = \frac{a}{b - 1}$
$N o w s i n^{2} (α + β) + a s i n (α + β) c o s (α + β) + b c o s^{2} (α + β) = c o s^{2} (α + β) [t a n^{2} (α + β) + a t a n (α + β) + b] = \frac{t a n^{2} (α + β) + a t a n (α + β) + b}{s e c^{2} (α + β)} = \frac{t a n^{2} (α + β) + a t a n (α + β) + b}{1 + t a n^{2} (α + β)}$
$= \frac{\frac{a^{2}}{(b - 1)^{2}} + a . \frac{a}{b - 1} + b}{1 + \frac{a^{2}}{(b - 1)^{2}}} = \frac{\frac{a}{b - 1} (\frac{a}{b - 1} + a) + b}{1 + \frac{a^{2}}{(b - 1)^{2}}} = \frac{\frac{a}{b - 1} (\frac{a + a b - a}{b - 1}) + b}{1 + \frac{a^{2}}{(b - 1)^{2}}} = b$

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Similar questions

If $t a n α, t a n β$ are the roots of the equation $x^{2}$ + ax + b = 0 then the value of $s i n^{2} (α + β) + a s i n (α + β) c o s (α + β) + b c o s^{2} (α + β) =$

Q.
If

tan α, tan β

are the roots of the equation

x^{2} + p x + q = 0 (p \neq 0)

then

{sin}^{2} (α + β) + p sin (α + β) cos (α + β) + q {cos}^{2} (α + β) =

tan α

and

tan β

are the roots of the equation

x^{2} + a x + b = 0

then the value of

{sin}^{2} (α + β) + a sin (α + β) cos (α + β) + b {cos}^{2} (α + β)

is equal to

Q. If

α, β

are the quadratic equation

x^{2} + a x + b = 0, (b \neq 0)

; then the quadratic equation whose roots are

α - \frac{1}{β}, β - \frac{1}{α}

Q. Statement I: lf

α, β

are the roots of

x^{2} - a x + b = 0

, then the equation whose roots are

\frac{α + β}{α}, \frac{α + β}{β}

b x^{2} - a^{2} x + a^{2} = 0

Statement II: lf

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are the roots of

x^{2} - b x + c = 0

and

α + h, β + h

are the roots of

x^{2} + q x + r = 0

, then

h = b - q

.
Which of the above statement(s) is(are) true.

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If $t a n α, t a n β$ are the roots of the equation $x^{2}$ + ax + b = 0 then the value of $s i n^{2} (α + β) + a s i n (α + β) c o s (α + β) + b c o s^{2} (α + β) =$