First slide

Equation of line in Straight lines

Question

The distance of origin from line $(1 + \sqrt{3}) y + (1 - \sqrt{3}) x = 10$ measured along the line $y = \sqrt{3} x + 4$ is

Moderate

A

$\frac{5}{\sqrt{2}}$
B

$5 \sqrt{2} + k$
C

10
D

5

Solution

Slope of the line $L \equiv (1 + \sqrt{3}) y + (1 - \sqrt{3}) x = 10$ is
$\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = \tan 15^{0}$
Slope of line $y = \sqrt{3} x + k$ is $\sqrt{3} = \tan 60^{0}$
In the figure, distance of origin from line L measured along the line $y = \sqrt{3} x + k$ is OP
From the figure,
$OP = OM \sec 45^{0}$
$= \frac{|0 + 0 - 10|}{{(\sqrt{3} - 1)}^{2} + {(\sqrt{3} + 1)}^{2}} \sqrt{2} = \frac{10}{\sqrt{8}} \sqrt{2} = 5$

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