Question

if the straight line drawn through the point P(2√3, 1) and making an angle $\pi /3$ with x axis, meet the line √3+y=2 at then length of PQ is
A. 10√3
B. 10/√3
C. 5/√3
D. 5√3

Answer 1

Answer:

sorry I have no answer

Answer 2

$\underline{\textsf{Given:}}$

$\textsf{The straight line drawn through the point P meets another line at q}$

$\underline{\textsf{To find:}}$

$\textsf{Length of PQ}$

$\underline{\textsf{Solution:}}$

$\textsf{Finding the line passes through P:}$

$\textsf{Slope,}\;\mathsf{m=tan\theta}$

$\textsf{Slope,}\;\mathsf{m=tan\dfrac{\pi}{3}=\sqrt{3}}$

$\textsf{Equation of the line is}$

$\mathsf{y-y_1=m(x-x_1)}$

$\mathsf{y-1=\sqrt{3}(x-2\sqrt{3})}$

$\mathsf{y-1=\sqrt{3}x-6}$

$\mathsf{\sqrt{3}x-y-5=0}$

$\textsf{This line meets}\;\mathsf{\sqrt{3}x+y-2=0}$

$\textsf{at Q}$

$\textsf{Finding Q by solving}$

$\mathsf{\sqrt{3}x-y-5=0}$

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

$\textsf{Adding,}\;\mathsf{2\sqrt{3}x-7=0}$

$\mathsf{x=\dfrac{7}{2\sqrt{3}}}$

$\textsf{From the second equation}$

$\mathsf{\sqrt{3}(\dfrac{7}{2\sqrt{3}})+y-2=0}$

$\mathsf{\dfrac{7}{2}+y-2=0}$

$\mathsf{\dfrac{7-4}{2}+y=0}$

$\mathsf{\dfrac{3}{2}+y=0}$

$\mathsf{y=-\dfrac{3}{2}}$

$\implies\mathsf{Q(\dfrac{7}{2\sqrt{3}},\,\dfrac{-3}{2})}$

$\textsf{Now,}$

$\textsf{Length of PQ}$

$\mathsf{=\sqrt{(2\sqrt{3}-\dfrac{7}{2\sqrt{3}})^2+(1+\dfrac{3}{2})}}$

$\mathsf{=\sqrt{(\dfrac{12-7}{2\sqrt{3}})^2+(\dfrac{2+3}{2})}}$

$\mathsf{=\sqrt{(\dfrac{5}{2\sqrt{3}})^2+(\dfrac{5}{2})}}$

$\mathsf{=\sqrt{\dfrac{25}{12}+\dfrac{25}{4}}}$

$\mathsf{=\sqrt{\dfrac{25+75}{12}}}$

$\mathsf{=\sqrt{\dfrac{100}{12}}}$

$\mathsf{=\sqrt{\dfrac{25}{3}}}$

$\mathsf{=\dfrac{\sqrt{25}}{\sqrt3}}$

$\implies\boxed{\mathsf{PQ=\dfrac{5}{\sqrt3}}}$

$\underline{\textsf{Answer:}}$

$\textsf{Option (C) is correct}$