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The equation of the circle with centre $(3, - 1)$ and which cuts off a chord of length 6 on the line $2 x - 5 y + 18 = 0$ is

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$x^{2} + y^{2} + 4 x - 16 y + 18 = 0$

$3 x^{2} + 3 y^{2} + 4 x - 6 y + 18 = 0$

$\begin{aligned} x^{2} + y^{2} + 4 x - 6 y + 8 = 0 \end{aligned}$

$\begin{aligned} x^{2} + y^{2} - 6 x + 2 y - 28 = 0 \end{aligned}$

answer is A.

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Detailed Solution

Let $' r$ ' be the radius of the circle.

$d =$ Distance from $(3, - 1) to 2 x - 5 y + 18 ≐ 0 = \frac{| 6 + 5 + 18 |}{\sqrt{4 + 25}} = \sqrt{29}$

Length of the chord $= 6 \Rightarrow 2 \sqrt{r^{2} - d^{2}} = 6 \Rightarrow r^{2} - 29 = 9 \Rightarrow r^{2} = 38$

$∴$ Circle equation is $(x - 3)^{2} + (y + 1)^{2} = 38 \Rightarrow x^{2} + y^{2} - 6 x + 2 y - 28 = 0 .$

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