First slide

Equation of line in Straight lines

Question

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 3 \sqrt{3})$ be three points. Then, the equation of the bisector of the angle $P Q R$ is

Moderate

A

$\frac{\sqrt{3}}{2} x + y = 0$
B

$x + \sqrt{3} y = 0$
C

$\sqrt{3} x + y = 0$
D

$x + \frac{\sqrt{3}}{2} y = 0$

Solution

The bisector of angle $P Q R$ divides $P R$ in. the ratio $P Q : Q R$ i.e. $1 : 6 .$ So, the coordinates of the point of division are
$(\frac{1 \times 3 + 6 \times - 1}{1 + 6}, \frac{1 \times 3 \sqrt{3} + 6 \times 0}{1 + 6}) = (\frac{- 3}{7}, \frac{3 \sqrt{3}}{7})$
Clearly, required bisector passes through $Q (0, 0)$ and $(- 3 / 7, 3 \sqrt{3} / 7) .$ So, its equation is
$y - 0 = \frac{\frac{3 \sqrt{3}}{7} - 0}{- \frac{3}{7} - 0} (x - 0) \Rightarrow y = - \sqrt{3} x \Rightarrow \sqrt{3} x + y = 0 .$

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