First slide

Equation of line in Straight lines

Question

P is a point on the line $y + 2 x = 1$ and , Q and R are two points on the line $3 y + 6 x = 6$ such that triangle PQR is an equilateral triangle. The length of the side of the triangle is

Moderate

A

$2 / \sqrt{5}$
B

$3 / \sqrt{5}$
C

$4 / \sqrt{5}$
D

None of these

Solution

The given lines are $y + 2 x = 1$ and $y + 2 x = 2$ are parallel lines
The distance between lines is $\frac{c_{2} - c_{1}}{\sqrt{a^{2} + b^{2}}} = \frac{2 - 1}{\sqrt{2^{2} + 1^{2}}} = \frac{1}{\sqrt{5}}$
The side length of the triangle is ‘s’
$s i n 60^{0} = \frac{p}{s} \frac{1}{2} = \frac{\frac{1}{\sqrt{5}}}{s} ∴ s = \frac{2}{\sqrt{5}}$

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