First slide

Pair of straight lines

Question

The distance between the pair of parallel lines $2 x^{2} + 4 xy + 2 y^{2} + 3 x + 3 y + 1 = 0$ is

Moderate

A

$\frac{1}{2 \sqrt{2}}$
B

$\frac{1}{\sqrt{2}}$
C

$\frac{1}{2}$
D

$\sqrt{2}$

Solution

The distance between the parallel lines ${ax}^{2} + 2 hxy + {by}^{2} + 2 gx + 2 fy + c = 0$ is $2 \sqrt{\frac{g^{2} - ac}{a (a + b)}}$
For the equation $2 x^{2} + 4 xy + 2 y^{2} + 3 x + 3 y + 1 = 0$ ,the values are $a = 2, b = 2, c = 1, g = \frac{3}{2}$
Hence the distance required is d
$\begin{matrix} d = 2 \sqrt{\frac{\frac{9}{4} - 2}{2 (4)}} \\ = 2 \sqrt{\frac{9 - 8}{2 (4) (4)}} \\ = 2 (\frac{1}{4 \sqrt{2}}) \\ = \frac{1}{2 \sqrt{2}} \end{matrix}$
Therefore, the required distance is $\frac{1}{2 \sqrt{2}}$

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