First slide

Pair of straight lines

Question

If the lines represented by the equations $2 x^{2} + mxy + y^{2} + 3 x + ly - 5 = 0$ are parallel to each other, which of the following is correct.

Easy

A

$l = \frac{3}{\sqrt{2}}, m = - 8$
B

$l = 3, m = 8$
C

$l = \pm \frac{3}{\sqrt{2}}, m = \pm 2 \sqrt{2}$
D

$l = 8, m = \frac{3}{\sqrt{2}}$

Solution

The conditions that the equation ${ax}^{2} + 2 hxy + {by}^{2} + 2 gx + 2 fy + c = 0$ represents pair of $h^{2} = ab$ and ${af}^{2} = {bg}^{2}$ parallel lines are and
For the equation $2 x^{2} + mxy + y^{2} + 3 x + ly - 5 = 0$ , the values are $a = 2, h = \frac{m}{2}, b = 1, g = \frac{3}{2}, f = \frac{l}{2}$
Hence,
$\begin{matrix} \frac{m^{2}}{4} = 2 \\ m^{2} = 8 \\ m = \pm 2 \sqrt{2} \end{matrix}$
And
$\begin{matrix} 2 {(\frac{l}{2})}^{2} = 1 {(\frac{3}{2})}^{2} \\ l^{2} = \frac{9}{2} \\ l = \pm \frac{3}{\sqrt{2}} \end{matrix}$
Therefore, $l = \pm \frac{3}{\sqrt{2}}, m = \pm 2 \sqrt{2}$

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