A square is inscribed in the circle $x^{2} + y^{2} - 2 x + 4 y - 93 = 0$ , with its sides parallel to the axes of co-ordinates. Which of the coordinates correspond to the vertices of the square.

(- 6, - 9)

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(- 6, - 5)

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(8, - 9)

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(8, - 5)

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Solution

The correct options are
A $(- 6, - 9)$
B $(8, - 9)$
Since, the square is inscribed in a circle, the vertices of the square will lie on the circle.
Hence, the points satisfying the equation of circle can be the coordinates of the square.
Points $(- 6, - 9)$ and $(8, - 9)$ can be the coordinates of the square.
The equation of the circle is
$x^{2} + y^{2} - 2 x + 4 y - 93 = 0$
Coordinates of the centre are $(1, - 2)$
Radius $= 7 \sqrt{2}$
When a square is inscribed in a circle, the diameter of the circle is equal to the length of the diagonal of the square.
Diameter $= 14 \sqrt{2}$
Side length=Distance between the points $(- 6, - 9)$ and $(8, - 9)$ is 14 units. The distance of the sides from the centre of the circle will be 7 units. The sides of the square are parallel to the coordinate axis. Hence,the equation of the sides will be $x = - 6$ , $x = 8$ , $y = - 9$ and $y = 5$
Hence, options 'A' and 'C' are correct.

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