A circle passes through $(- 2, 4)$ and touches the $y -$ axis at $(0, 2) .$ Then which of the following equations can represent a diameter of the circle?

4 x + 5 y - 6 = 0

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2 x - 3 y + 10 = 0

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3 x + 4 y - 3 = 0

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5 x + 2 y + 4 = 0

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Solution

The correct option is B $2 x - 3 y + 10 = 0$

Let centre of the circle be $(h, k)$
As it touches $y -$ axis at $(0, 2),$ then
$k = 2$ and $r = | h |$

Also, $(- 2, 4)$ lies on the given circle, so distance between the point and centre is equal to radius.
$\Rightarrow \sqrt{(h + 2)^{2} + (2 - 4)^{2}} = r \Rightarrow h^{2} + 4 + 4 h + 4 = h^{2} \Rightarrow h = - 2$
So, centre is $(- 2, 2),$ which lies on the line $2 x - 3 y + 10 = 0$
Hence, $2 x - 3 y + 10 = 0$ is one of the diameters of the circle.

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