The joint equation of bisectors of angles between lines $x = 5, y = 3$ is .......

(x - 5) (y - 3) = 0

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x^{2} - y^{2} - 10 x + 6 y + 16 = 0

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x y = 0

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x y - 5 x - 3 y + 15 = 0

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Solution

The correct option is D $x^{2} - y^{2} - 10 x + 6 y + 16 = 0$
Equation of the bisectors of angles between the lines $x = 5$ , $y = 3$ is
$(y - y_{1}) = m (x - x_{1})$
where $x_{1} = 5, y_{1} = 3$
and the lines are parallel to the x and y axes so the bisector of angle makes $45^{\circ}$
$∴$ $m = \pm 1$
$(y - 3) - (x - 5) = 0$ ------(1)
$(y - 3) + (x - 5) + 0$ ------(2)
Multiplying both equations gives
$x^{2} - y^{2} + 6 y - 10 x + 16 = 0$

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