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(A) The general equation of a pair of lines passing through the origin is given by $Ax^2 + 2Hxy + By^2 = 0$. For these lines to be at right angles to

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$2$

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(A) The general equation of a pair of lines passing through the origin is given by $Ax^2 + 2Hxy + By^2 = 0$. For these lines to be at right angles to each other,the sum of the coefficients of $x^2$ and $y^2$ must be zero,i.e.,$A + B = 0$. In the given equation $3ax^2 + 5xy + (a^2 - 2)y^2 = 0$,we have $A = 3a$ and $B = a^2 - 2$. Setting $A + B = 0$,we get $3a + a^2 - 2 = 0$,which is $a^2 + 3a - 2 = 0$. Solving this quadratic equation for $a$ using the quadratic formula $a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$,we get $a = \frac{-3 \pm \sqrt{9 - 4(1)(-2)}}{2} = \frac{-3 \pm \sqrt{17}}{2}$. Since the discriminant $D = 17 > 0$,there are two distinct real values for $a$. Thus,the number of values of $a$ is $2$.

The number of values of $a$ for which the pair of lines represented by $3ax^2 + 5xy + (a^2 - 2)y^2 = 0$ are at right angles to each other,is

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