The point of lines represented by $3 a x^2+5 x y+\left(a^2-2\right) y^2=0$ and perpendicular to each other for

The point of lines represented by $3 a x^2+5 x y+\left(a^2-2\right) y^2=0$ and perpendicular to each other for
  1. two values of a
  2. $\forall a$
  3. for one value of a
  4. for no values of a

Solution

$3 a+a^2-2=0 \Rightarrow a^2+3 a-2=0 \Rightarrow a=\frac{-3 \pm \sqrt{9+8}}{2}=\frac{-3 \pm \sqrt{17}}{2}$

Asked in: JEE Main 2002

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