Question

If $p$ be the length of the perpendicular from the origin on the straight line $a x+b y=p$ 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 $a x+b y-p=0$,
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} $
$\Rightarrow a =\frac{1}{2}$
$\left[ a =-\frac{1}{2}\right.$ 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 $=60^{\circ}$