The equation to the circle which is touched by $y = x$ , has its centre on the positive direction of the x axis and cuts off a chord of length $2$ units along the line $y - \frac{x}{\sqrt{3}} = 0$

x^{2} + y^{2} - 4 x + 2 = 0

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x^{2} + y^{2} - 4 x + 1 = 0

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x^{2} + y^{2} - 8 x + 8 = 0

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x^{2} + y^{2} - 4 y + 2 = 0

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Solution

The correct option is A $x^{2} + y^{2} - 4 x + 2 = 0$
Choice (B) is ruled out since $y = x$ does not make it perfect square.
Choice (D) is ruled out since its centre lies on the y-axis
Choice (A) $y = x$ is a tangent to the circle. Perpendicular distance from $(2, 0)$ on $y - \frac{x}{\sqrt{3}} = 0$ is $∣ ∣ ∣ ∣ ∣ ∣ \frac{- \frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}}} ∣ ∣ ∣ ∣ ∣ ∣$
and the radius of the circle is $\sqrt{4 - 2} = \sqrt{2}$ and hence the length of the chord is $2 \sqrt{2 - 1} = 2$ .
Hence choice (A) is correct.

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The equation to the circle which is touched by y=x, has its centre on the positive direction of the x axis and cuts off a chord of length 2 units along the line y−x√3=0

The equation to the circle which is touched by $y = x$ , has its centre on the positive direction of the x axis and cuts off a chord of length $2$ units along the line $y - \frac{x}{\sqrt{3}} = 0$