lf $x^{2} + α y^{2} + 2 β y = a^{2}$ represents a pair of perpendicular straight lines, then

α = 1, β = a

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α = 1, β = - a

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α = - 1, β = - a

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α = - 1, β = a

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Solution

The correct options are
C $α = - 1, β = - a$
D $α = - 1, β = a$
$x^{2} + α y^{2} + 2 β y = a^{2}$ represent pair of perpendicular straight lines
Coeff. of $x^{2} +$ Coeff. of $y^{2} = 0$
$1 + α = 0$
$α = - 1$
and Discriminant $= D = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
$= - α a^{2} - β^{2} = 0$
$α a^{2} + β^{2} = 0$
and $α = - 1, β^{2} = a^{2}$
$α - 1$ and $β = a$ or $β = - a$
Answer is $C$ & $D .$

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