First slide

Equation of line in Straight lines

Question

$Distance of the origin from the line (1 + \sqrt{3}) y + (1 - \sqrt{3}) x = 10 along the line y = \sqrt{3} x + k is$

Easy

A

$5 / \sqrt{2}$
B

$5 \sqrt{2} + k$
C

10
D

0

Solution

$T h e d i s \tan c e f r o m a p o i n t P (x_{1}, y_{1}) t o a l i n e a x + b y + c = 0 a l o n g a l i n e w h i c h m a k e s a n a n g l e θ$
$w i t h x - a x i s t h e n d = |\frac{a x_{1} + b y_{1} + c}{a \cos θ + b \sin θ}|$

$T h e d i s \tan c e f r o m o r i g i n t o t h e l i n e (1 + \sqrt{3}) y + (1 - \sqrt{3}) x = 10 a l o n g a l i n e w h o s e i n c l i n a t i o n i s 60^{\circ}$
$d = |\frac{10}{(1 - \sqrt{3}) \cos 60^{\circ} + (1 + \sqrt{3}) \sin 60^{\circ}}| = |\frac{10}{\frac{1}{2} - \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} + \frac{3}{2}}| = |\frac{10}{2}| = 5$

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