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 lengt

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

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1 month 2021-08-11T14:36:37+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-11T14:37:39+00:00

    Answer:

    sorry I have no answer

  1. Ava
    0
    2021-08-11T14:38:03+00:00

    \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}

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