Question

Let $a, b \in R$ be such that the equation $a x^{2}-2 b x+15=0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-2 b x+21=0$, then $\alpha^{2}+\beta^{2}$ is equal to:

• A. 37
• B. 58
• C. 68
• D. 92

Answer 1

Answer: (B)

$ax ^{2}-2 bx +15=0$
$2 \alpha=\frac{2 b }{ a }, \alpha^{2}=\frac{15}{ a }$
$\frac{\alpha}{2}=\frac{15}{2 b }$
$\alpha=\frac{15}{ b }$
$x ^{2}-2 bx +21=0$
$\left(\frac{15}{ b }\right)^{2}-2 b \left(\frac{15}{ b }\right)+21=0$
$b ^{2}=25$
$\alpha+\beta=2 b , \alpha \beta=21$
$\alpha^{2}+\beta^{2}=4 b ^{2}-42$
$=58$