Question

A straight line makes an angle \[\dfrac{\pi }{3}\] with positive direction of x-axis measured in anti clock direction and makes positive intercept on y-axis and the line is at a perpendicular distance of 5 units from origin then its equation is?
\[(A)\text{ }x+\sqrt{3}y=10\]
\[(B)\text{ }x=\sqrt{3}y+10\]
\[(C)\text{ y}=\sqrt{3}x+10\]
\[(D)\text{ y}=\sqrt{3}x-10\]

Answer 1

Hint: Here we are given that straight line makes angle with positive direction of x axis and perpendicular distance from origin is also given, so we will use the equation of line in normal form to find equation of line. Equation of line in normal form is given by \[x\cos \varphi +y\cos \varphi =P\] where \[\varphi \] is the angle formed by line with positive direction of x axis, and P is perpendicular distance from the origin to the given line.

Complete step by step answer:
We are given a perpendicular distance of line from origin as 5 units. So, we can write P=5.
Also, we are given that the line forms an angle \[\dfrac{\pi }{3}\] with positive direction of x axis. So, we can write
\[\varphi =\dfrac{\pi }{3}\].
We know the equation of line in normal form is given by \[x\cos \varphi +y\cos \varphi =P\], where is angle \[\varphi \]formed by the line with positive direction of x axis and P is the perpendicular distance from line to the origin.
Putting values in the equation, we get
\[\Rightarrow x\cos \dfrac{\pi }{3}+y\sin \dfrac{\pi }{3}=P\]
\[\Rightarrow x(\dfrac{1}{2})+y(\dfrac{\sqrt{3}}{2})=5\]
Taking \[\dfrac{1}{2}\] common from left side, we get
\[\Rightarrow \dfrac{1}{2}[x+\sqrt{3}y]=5\]
Multiplying both side by 2, we get
\[\Rightarrow x+\sqrt{3}y=10\]

So, the correct answer is “Option A”.

Note: Students should learn this formula for solving sum easily. They should always check that a given angle should be made between the line and positive direction of the x axis only. Also, perpendicular distance should be taken from origin only otherwise the formula is not applicable.