The locus of the image of origin in line rotating about the point $(1, 1)$ is

x^{2} + y^{2} = 2 (x + y)

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x^{2} + y^{2} = (x + y)

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x^{2} + y^{2} = 2 (x - y)

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x^{2} + y^{2} = (x - y)

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Solution

The correct option is D $x^{2} + y^{2} = 2 (x + y)$
Given that a line is rotated about $(1, 1)$ Hence if we rotate line about a point a circle is formed
So here a circle is formed whose centre be $(1, 1)$ whose radius will be distance from origin to point $(1, 1)$ becuase we see the image of origin
$r = \sqrt{(1)^{2} + (1)^{2}} = \sqrt{2}$
Hence equation of circle from centre $(1, 1)$ and radius $r$
be the locus of image of origin
$(x - 1)^{2} + (y - 1)^{2} = (\sqrt{2})^{2}$

$x^{2} + 1 - 2 x + y^{2} + 1 - 2 y = 2$

$x^{2} + y^{2} = 2 - 2 + 2 (x + y)$
$x^{2} + y^{2} = 2 (x + y)$

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The locus of the image of origin in line rotating about the point (1,1) is

The locus of the image of origin in line rotating about the point $(1, 1)$ is