First slide

Pair of straight lines

Question

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

Moderate

A

$2 \sqrt{5} a$
B

$10 \sqrt{a}$
C

$10 a$
D

$5 \sqrt{2} a$

Solution

We have,
$\begin{array}{l} x^{2} + y^{2} + 2 x y - 8 a x - 8 a y - 9 a^{2} = 0 \\ \Rightarrow (x + y)^{2} - 8 a (x + y) - 9 a^{2} = 0 \\ \Rightarrow (x + y - 9 a) (x + y + a) = 0 \\ \Rightarrow x + y - 9 a = 0, x + y + a = 0 \end{array}$
Clearly, these lines are parallel. The distanced between these lines is
$d = \frac{| a - (- 9 a) |}{\sqrt{1^{2} + 1^{2}}} = 5 \sqrt{2} a$

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