niteshwaghray wrote:

In a rectangular coordinate plane, AB is the diameter of a circle and point C lies on the circle. If the coordinates of points A and B are (-1,0) and (5,0), and the area of triangle ABC is \(6\sqrt{2}\)square units, which of the following can be the coordinates of point C?

A. (0, 2\(\sqrt{2}\))

B. (1, 2\(\sqrt{2}\))

C. (\(\sqrt{2}\),2)

D. (2,\(\sqrt{2}\))

E. (2\(\sqrt{2}\),1)

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We have \(AB=6 \implies R = 3\).

\(I\) is the center of that circle, then we have \(I(2,0)\)

The equation of that circle is \((I): \; \; (x-2)^2 + y^2 = 3^2\)

The coordinates of \(C(x_C, y_C)\). We have \((x_C-2)^2 + y_C^2 = 9\)

Also, we have

\(S_{ABC}=6\sqrt{2} \implies \frac{AB \times |y_C|}{2}=6\sqrt{2} \\

\implies |y_C|=2\sqrt{2} \implies y_C^2=8 \)

\(\implies (x_C-2)^2 = 1 \implies x_C = 3\) or \(x_C = 1\).

Only choice B fits the roots. B is the correct answer.

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