• # question_answer The pair of lines represented by $3a{{x}^{2}}+5xy+({{a}^{2}}-2){{y}^{2}}=0$ are perpendicular to each other for   [AIEEE 2002] A)            Two values of $a$            B)            $\forall a$ C)            For one value of $a$       D)            For no value of $a$

$\because$ The lines are perpendicular, if            coefficient of ${{x}^{2}}$ + coefficient of ${{y}^{2}}=0$            Þ $3a+({{a}^{2}}-2)=0$ Þ ${{a}^{2}}+3a-2=0$            $\because$ The equation is a quadratic equation in ?a? and ${{B}^{2}}-4AC>0$.            \ The roots of a are real and distinct. Therefore, the lines are perpendicular to each other for two values of ?a?.