Question

If the straight line through the point P(3, 4) makes an angle $\frac{\pi }{6}$ with the x–axis and meets the line 12x + 5y + 10 = 0 at Q then the length PQ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

The equation of any time passing through the given P(3, 4) and making an angle $\frac{\pi }{6}$ with x-axis is $\frac{x-3}{cos{30}^{\circ }}=\frac{y-4}{sin{30}^{\circ }}=r$ (say) .....(i)
Where ‘r’ represents the distance of any point Q on this line from the given point P(3, 4)
The coordinates (x, y) of any point Q on line (i) are $(3+rcos{30}^{\circ },4+rsin{30}^{\circ })$ i.e., $(3+\frac{r\sqrt{3}}{2},4+\frac{r}{2})$
If the point lies on the line 12x + 5y + 10 = 0, then
$12(3+\frac{r\sqrt{3}}{2})+5(4+\frac{r\sqrt{3}}{2})+10=0\Rightarrow r=\frac{132}{12\sqrt{3}+5}$