First slide

Theory of equations

Question

If the equation ${ax}^{2} + 2 bx + 3 c = 0$ and $3 x^{2} + 8 x + 15 = 0$ have a common root, where a,band c are the lengths of the sides of a $△ ABC,$ then $\sin^{2} A + \sin^{2} B + \sin^{2} C$ is equal to

Moderate

A

1
B

$\frac{3}{2}$
C

$\sqrt{2}$
D

2

Solution

Discriminant o{ the equation $3 x^{2} + 8 x + 15 = 0$ is given by
$D = 64 - 180 = - 116 < 0$
so it , roots are imaginary and therefore roots are conjugate to each other. Therefore, one common root means both the roots are common.
$∴ \frac{a}{3} = \frac{2 b}{8} = \frac{3 c}{15}$
$\Rightarrow \frac{a}{3} = \frac{b}{4} = \frac{c}{5} = k (say), k \neq 0$
Now, a² + b² = c²
$\Rightarrow ΔABC isrightangled .$
$\begin{array}{l} ∴ \sin^{2} A + \sin^{2} B = \sin^{2} C \\ \Rightarrow \sin^{2} A + \sin^{2} B + \sin^{2} C = 2 \sin^{2} C = 2 \sin^{2} 90^{\circ} = 2 \end{array}$

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