The pair of lines represented by 3ax^2 + 5xy | vedclass

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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 perpendicular,the co

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Two values of $a$

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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 perpendicular,the condition is that 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 simplifies to $a^2 + 3a - 2 = 0$.This is a quadratic equation in $a$. The discriminant $D = b^2 - 4ac = (3)^2 - 4(1)(-2) = 9 + 8 = 17$.Since $D > 0$,the quadratic equation $a^2 + 3a - 2 = 0$ has two distinct real roots for $a$.Therefore,the lines are perpendicular to each other for two values of $a$.

The pair of lines represented by $3ax^2 + 5xy + (a^2 - 2)y^2 = 0$ are perpendicular to each other for

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