The distance between the pair of parallel lines $x^{2} + 2 x y + y^{2} - 8 a x - 8 a y - 9 a^{2} = 0$ is

2 \sqrt{5} a

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\sqrt{10} a

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10 a

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5 \sqrt{2} a

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Solution

The correct option is D $5 \sqrt{2} a$
Given Eq
$x^{2} + 2 x y + y^{2} - 8 a x - 8 a y - 9 a^{2} = 0$
$x = \frac{- (2 y - 8 a) \pm \sqrt{64 a^{2} - 32 a y + 4 y^{2} - 4 (y^{2} - 8 a y - 9 a^{2})}}{2}$
$2 x = - (2 y - 8 a) \pm \sqrt{64 a^{2} - 32 a y + 4 y^{2} - 4 y^{2} + 32 a y + 36 a^{2})}$
$2 x = - (2 y - 8 a) \pm \sqrt{100 a^{2}}$
$2 x = - (2 y - 8 a) \pm 10 a$
$2 x + 2 y = 18 a$ and $2 x + 2 y = - 2 a$
$2 x + 2 y - 18 a = 0$ and $2 x + 2 y + 2 a = 0$
Here lines are parallel Distance between $2 x + 2 y - 18 a = 0$ and $2 x + 2 y + 2 a = 0$ by formula is
$d i s t a n c e = ∣ ∣ ∣ \frac{c_{1} - c_{2}}{\sqrt{a^{2} + b^{2}}} ∣ ∣ ∣$
$d i s t a n c e = ∣ ∣ ∣ ∣ \frac{- 18 a - 2 a}{\sqrt{2^{2} + 2^{2}}} ∣ ∣ ∣ ∣$
$d i s t a n c e = ∣ ∣ ∣ \frac{- 20 a}{\sqrt{8}} ∣ ∣ ∣$
$d i s t a n c e = \frac{20 a \sqrt{8}}{8}$
$d i s t a n c e = 5 \sqrt{2} a$

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