If $ x^{2}+\alpha y^{2}+2 \beta y=a^{2} $ represents a pair of perpendicular lines, then $ \beta $ equals to

Question

If $ x^{2}+\alpha y^{2}+2 \beta y=a^{2} $ represents a pair of perpendicular lines, then $ \beta $ equals to, $ 4 a $, a, $ 2 \mathrm{a} $, $ 3 a $

MARKS App · Accepted Answer

Let given line be x 2 + α y 2 + 2 β - a 2 = 0 Here, a = 1 , b = α , h = 0 , g = 0 , f = β , c = - a 2 Condition for perpendicular line a + b = 0 ∴ 1 + α = 0 ⇒ = - 1 Condition of pair of lines a b c + 2 f g h - a f 2 - b g 2 - c h 2 = 0 ∴ 1 × α × - a 2 + 0 - 1 β 2 - 0 - - a 2 0 = 0 ⇒ - a 2 α - β 2 = 0 ⇒ β 2 = - α a 2 ⇒ β 2 = - - 1 a 2 ∵ α = - 1 ⇒ β 2 = a 2 ⇒ β = a

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