Question

If $p$ be the length of the perpendicular from the origin on the straight line $ax+by=p$ and $b=\frac{\sqrt{3}}{2}$, then what is the angle between the perpendicular and the positive direction of $x-$ axis?

• A. $30^{\circ }$
• B. $45^{\circ }$
• C. $60^{\circ }$
• D. $90^{\circ }$

Answer 1

Answer: (C)

Equation of line is $ax+by-p=0$,
then length of perpendicular, from the origin is
$p=\left|\frac{a\times 0+b\times 0-p}{\sqrt{a^2+b^2}}\right|$ or
$p=\left|\frac{-p}{\sqrt{a^2+b^2}}\right|$
$\Rightarrow a^2+b^2=1$
$b=\frac{\sqrt{3}}{2}\text{ or }{b}^2=\frac{3}{4}$
$\Rightarrow a^2+\frac{3}{4}=1$
$\Rightarrow a^2=\frac{1}{4}$
$\Rightarrow a=\frac{1}{2}$
$[a=-\frac{1}{2}$ not taken since angle is with $+$ vedirection to $x$ -axis.]
Equation is $\frac{1}{2}x+\frac{\sqrt{3}}{2}y=p$ or
$x\cos 60^{\circ }+y\sin 60^{\circ }=p$
Angle $=60^{\circ }$