Question

Equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of $120°$with the x-axis, is

• A. $x\surd 3+y+8=0$
• B. $x\surd 3+y=8$
• C. $x\surd 3–y=8$
• D. $x–\surd 3y+8=0$

Answer 1

The correct option is B

$x\surd 3+y=8$

Explanation for the correct option:

Step 1. Find $\angle MAO$

Length of the perpendicular (OM) $=4$

$\angle BAX=120°$

$\Rightarrow$$\angle BAO=180°-120°$ [$\because$Sum of Linear pair angles $=180°$]

$=60°$

$\therefore \angle MAO=60°$

Step 2. In $\Delta AMO$, we have

$\angle MAO+\angle AOM+\angle OMA=180°$ [By angle sum property of triangle]

$\Rightarrow$ $60°+\theta +90°=180°$

$\Rightarrow$ $150°+\theta =180°$

$\Rightarrow$ $\theta =180°-150°$

$\Rightarrow$ $\theta =30°$

$\therefore \angle AOM=30°$

Step 3. Find the equation in normal form

$x\cos \theta +y\sin \theta =p$

$\Rightarrow$$x\cos (30°)+y\sin (30°)=4$ $\left[\because p=4\right]$

$\Rightarrow$ $x\left(\frac{\sqrt{3}}{2}\right)+y\left(\frac{1}{2}\right)=4$

$\Rightarrow$ $\surd 3x+y=8$

Thus, the required equation is $\mathbf{\surd }\mathbf{3}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}$

Hence, Option ‘B’ is Correct.