The equation of the circle with the chord $y = 2 x$ of the circle $x^{2} + y^{2} - 10 x = 0$ as its diameter is

x^{2} + y^{2} - 2 x - 4 y - 5 = 0

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

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

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none of these

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Solution

The correct option is B $x^{2} + y^{2} = 2 x + 4 y$
Since $y = 2 x$ is a chord of the circle

$x^{2} + y^{2} - 10 x = 0$

$\Rightarrow x^{2} + 4 x^{2} - 10 x = 0$

$\Rightarrow x = 0, 2$

$\Rightarrow y = 0, 4$

So, the end -points of diameter are $(0, 0)$ and $(2, 4)$

Equation of circle whose end -points of diameter are $(0, 0)$ and $(2, 4)$ is

$(x - 0) (x - 2) + (y - 0) (y - 4) = 0$

$\Rightarrow x^{2} + y^{2} - 2 x - 4 y = 0$

$\Rightarrow x^{2} + y^{2} = 2 x + 4 y$

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The equation of the circle with the chord y=2x of the circle x2+y2−10x=0 as its diameter is

The equation of the circle with the chord $y = 2 x$ of the circle $x^{2} + y^{2} - 10 x = 0$ as its diameter is