Question

The pair of lines represented by $3a{x}^2+5xy+(a^2-2)y^2=0$ and at right angles to each other, then value $\left(s\right)$ of $a$ is/are:

• A. $\frac{-3+\sqrt{17}}{2}$
• B. $\frac{-3-\sqrt{17}}{2}$
• C. $\frac{3+\sqrt{17}}{2}$
• D. $\frac{3-\sqrt{17}}{2}$

Answer 1

Answer: (A), (B)

The correct options are
A $\frac{-3+\sqrt{17}}{2}$
B $\frac{-3-\sqrt{17}}{2}$

The pair of lines represented by $3a{x}^2+5xy+(a^2-2)y^2=0$ and at right angles to each other.
pair of lines given by $a{x}^2+2hxy+b{y}^2=0$ are at right angles if $a+b=0$
$\Rightarrow 3a+a^2-2=0$
$\Rightarrow a^2+3a-2=0$
$\therefore a=\frac{-3\pm \sqrt{17}}{2}$
Hence, options A and B.