First slide

Pair of straight lines

Question

The combined equation of the angle bisectors of the lines represented by $2 x^{2} + x y - y^{2} - 10 x + 2 y + 8 = 0$ is

Moderate

A

$x^{2} + 6 x y - y^{2} - 8 x - 16 y + 24 = 0$
B

$x^{2} - 6 x y - y^{2} + 8 x - 16 y + 40 = 0$
C

$x^{2} - 6 x y - y^{2} + 8 x + 16 y - 24 = 0$
D

$x^{2} + 6 x y - y^{2} - 8 x + 16 y - 40 = 0$

Solution

$a = 2, 2 h = 1, b = - 1, 2 g = - 10, 2 f = 2, c = 8$
Point of intersection is $(x_{1}, y_{1}) = (\frac{h f - b g}{a b - h^{2}}, \frac{g h - a f}{a b - h^{2}}) = (2, 2)$
The pair of angular bisectors is $h ({(x - x_{1})}^{2} - {(y - y_{1})}^{2}) = (a - b) (x - x_{1}) (y - y_{1})$
$\Rightarrow \frac{1}{2} ({(x - 2)}^{2} - {(y - 2)}^{2}) = (2 + 1) (x - 2) (y - 2)$
$\Rightarrow x^{2} - y^{2} - 4 x + 4 y = 6 (x y - 2 x - 2 y + 4)$
$\Rightarrow x^{2} - 6 x y - y^{2} + 8 x + 16 y - 24 = 0$

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