Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line x + y = 1. If the intercepts made by

the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equations can represent $L_{1}$

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Solution

The correct option is B

$x + y = 1, x^{2} + y^{2} - x + 3 y = 0 x^{2} + (1 - x)^{2} - x + 3 (1 - x) = 0 x^{2} + 1 + x^{2} - 2 x - x + 3 - 3 x = 0 2 x^{2} - 6 x + 4 = 0 x^{2} - 3 x + 2 = 0 x = 1, 2 A (1, 0), B (2, - 1) A B = \sqrt{1 + 1} = \sqrt{2} ∴ y = x 2 x^{2} + 2 x = 0 x = 0, - 1 y = x (0, 0), (- 1, - 1) \sqrt{1 + 1} = \sqrt{2}$

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