First slide

Equation of line in Straight lines

Question

If the straight line drawn through the point $P (\sqrt{3}, 2)$ and making an angle $\frac{π}{6}$ with the x-axis meets the line $\sqrt{3} x - 4 y + 8 = 0$ at Q, then the length of PQ is

Moderate

A

4
B

5
C

6
D

none of these

Solution

The given line is
$\sqrt{3} x - 4 y + 8 = 0$ ---(1)
Let PQ = r
Then, the coordinates of Q are
$(\sqrt{3} + r \cos \frac{π}{6}, 2 + r \sin \frac{π}{6}) o r (\sqrt{3} + \frac{\sqrt{3}}{2} r, 2 + \frac{r}{2})$
Since the point Q lies on the given line,
$∴ \sqrt{3} (\sqrt{3} + \frac{\sqrt{3}}{2} r) - 4 (2 + \frac{r}{2}) + 8 = 0$
$\Rightarrow 6 + 3 r - 16 - 4 r + 16 = 0 o r r = 6$
Hence, PQ = 6

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