Question

lf the equations $a{x}^2+2bx+3c=0$ and $3x^2+8x+15=0$ have a common root, where $a,b,c$ are the length of the sides of a $\Delta ABC$, then ${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C$ is equal to

• A. $1$
• B. $\frac{3}{2}$
• C. $\sqrt{2}$
• D. $2$

Answer 1

The correct option is D $2$

Equations $a{x}^2+2bx+3c=0$ and $3x^2+8x+15=0$ have a common root.

The second equation has purely imaginary roots which occur in conjugate pairs. The first equation has real coefficients since it is given that they are sides of a triangle.

Therefore, both the roots of the equation are common
$\Rightarrow \frac{a}{3}=\frac{2b}{8}=\frac{3c}{15.}$
$\therefore a,b,c$ are $\{3,4,5\}\Rightarrow$ right angled triangle
$\therefore {\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C=1+{\sin }^{2}B+{\cos }^{2}B$
$=1+1$
$=2.$