Mathematics

# If the straight line through the point $P(3,4)$ makes an angle $\dfrac{\pi}{6}$ with the $x-$axis and meets the line $12x+5y+10=0$ at $Q$, find the length of $PQ$.

##### SOLUTION
Any line through $P(3,4)$ making an angle  $\dfrac{\pi}{6}$ with $x-$axis is
$\dfrac { x-3 }{ \cos { { 30 }^{ o } } } =\dfrac { y-4 }{ \sin { { 30 }^{ o } } } =r$
Where $r$ represents the distance of any point on this line from the given point $P(3,4)$
Any point $Q$ on it is ${(r\sqrt {3}/2)+3,(r/2)+4}$ and $Q$ lies on $12x+5y+10=0$
$\therefore\quad 12\left[ \left( \dfrac { r\surd 3 }{ 2 } \right) +3 \right] +5\left( \dfrac { r }{ 2 } +4 \right) +10=0$
$\therefore\quad (12\sqrt {3}+5)r+132=0$
$\therefore\quad r=\dfrac { \left| -132 \right| }{ 12\surd \left( 3 \right) +5 } =\dfrac { 132 }{ 12\surd \left( 3 \right) +5 } .$

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Subjective Medium Published on 09th 09, 2020
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