First slide

Definition of a circle

Question

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

Moderate

A

$(- 6, - 9), (- 6, 5), (8, - 9), (8, 5)$
B

$(- 6, 9), (- 6, - 5), (8, - 9), (8, 5)$
C

$(- 6, - 9), (- 6, 5), (8, 9), (8, 5)$
D

$(- 6, - 9), (- 6, 5), (8, - 9), (8, - 5)$

Solution

Let $x = a, x = b, y = c, y = d$ be the sides of the square. The length of each diagonal of the square is equal to the diameter of the circle
$∴ Diagonal = 2 \sqrt{1 + 4 + 93} = 14 \sqrt{2}$
Let $I$ be the length of each side of the square. Then,
$2 I^{2} = (Diagonal)^{2} \Rightarrow 2 l^{2} = (14 \sqrt{2})^{2} \Rightarrow l = 14$ .
Therefore, each side of the square is at a distance 7 from the centre $(1, - 2)$ of the given circle. This implies that
$a = - 6, b = 8, c = - 9, d = 5$
Hence, the vertices of the square are (-6,-9), (-6,5), (8,-9), (8, 5).

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