First slide

Pair of straight lines

Question

The equation to the pair of lines parallel to $2 x^{2} + 4 xy + y^{2} = 0$ and passing through $(1, 2)$ is $2 x^{2} + 4 xy + y^{2} + kx - 8 y + 14 = 0$ then k=

Moderate

A

-8
B

-11
C

-12
D

8

Solution

The combined equation of pair of lines passing through the point $(x_{1}, y_{1})$ and parallel to the pair of lines ${ax}^{2} + 2 hxy + {by}^{2} = 0$ is $a {(x - x_{1})}^{2} + 2 h (x - x_{1}) (y - y_{1}) + b {(y - y_{1})}^{2} = 0$
Hence, the required equation is $2 {(x - 1)}^{2} + 4 (x - 1) (y - 2) + 2 {(y - 2)}^{2} = 0$
Simplify,
$\begin{matrix} 2 {(x - 1)}^{2} + 4 (x - 1) (y - 2) + {(y - 2)}^{2} = 0 \\ 2 x^{2} - 4 x + 2 + 4 xy - 8 x - 4 y + 8 + y^{2} - 4 y + 4 = 0 \\ 2 x^{2} + 4 xy + y^{2} - 12 x - 8 y + 14 = 0 \end{matrix}$
The coefficient of x in the above equation is $k = - 12$

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