First slide

Theory of equations

Question

If sin q and cos q are the roots of the equation ax² +bx + c = 0, then

Moderate

A

$(a - c)^{2} = b^{2} - c^{2}$
B

$(a - c)^{2} = b^{2} + c^{2}$
C

$(a + c)^{2} = b^{2} - c^{2}$
D

$(a + c)^{2} = b^{2} + c^{2}$

Solution

Since sin q and cos q are the roots of the equation ax² +bx + c = 0
$\begin{array}{ll} ∴ & \sin θ + \cos θ = - \frac{b}{a} and \sin θcos θ = \frac{c}{a} \\ Now (\sin θ + \cos θ)^{2} = 1 + 2 \sin θcos θ & ∴ \frac{b^{2}}{a^{2}} = 1 + \frac{2 c}{a} = \frac{a + 2 c}{a} \\ ∴ & b^{2} = a (a + 2 c) = a^{2} + 2 ac \\ \Rightarrow & b^{2} + c^{2} = a^{2} + 2 ac + c^{2} = (a + c)^{2} \\ \Rightarrow & (a + c)^{2} = b^{2} + c^{2} \end{array}$

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