Let B and C be the two points on the line y + x = 0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y - 2 x = 2 such that A B C is an equilateral triangle. Then, the area of A B C is [2023]

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Solution

Q.
Let B and C be the two points on the line $y + x = 0$ such that B and C are symmetric with respect to the origin. Suppose A is a point on $y - 2 x = 2$ such that $∆ A B C$ is an equilateral triangle. Then, the area of $∆ A B C$ is [2023]

1 $2 \sqrt{3}$

2 $\frac{8}{\sqrt{3}}$

3 $3 \sqrt{3}$

4 $\frac{10}{\sqrt{3}}$

Ans.
(2)

$Let point x = α lie on line y - 2 x = 2 and point x = β lie on line y + x = 0$

$Now, slope of A M = \frac{2 + 2 α}{α} = 1$

$\Rightarrow α = - 2$

$A M = 2 \sqrt{2}$

$And β = \frac{2}{\sqrt{3}}$

$A B = 2 \sqrt{2} \times \frac{2}{\sqrt{3}} = \frac{4 \sqrt{2}}{\sqrt{3}}$

$Area of ∆ A B C = \frac{\sqrt{3}}{4} {(A B)}^{2} = \frac{\sqrt{3}}{4} \times \frac{32}{3} = \frac{8}{\sqrt{3}}$

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