A circle is drawn through the points (3, 2), (-1, -2) and (5, -4). Find the :
(a) coordinates of the centre of the circle ;
(b) radius of the circle ;
(c) equation of the circle.
\((x - a)^{2} + (y - b)^{2} = r^{2}\)
\(x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}\)
Using A(3, 2) :
\(9 - 6a + a^{2} + 4 - 4b + b^{2} = r^{2}\)
\(13 - 6a - 4b + a^{2} + b^{2} = r^{2} .... (1)\)
Using B(-1, -2) :
\(1 + 2a + a^{2} + 4 + 4b + b^{2} = r^{2}\)
\(5 + 2a + 4b + a^{2} + b^{2} = r^{2} .... (2)\)
Using C(5, -4) :
\(25 - 10a + a^{2} + 16 + 8b + b^{2} = r^{2}\)
\(41 - 10a + 8b + a^{2} + b^{2} = r^{2} ... (3)\)
(1) - (2) :
\(8 - 8a - 8b = 0 \implies 1 - a - b = 0 .... (4)\)
(3) - (2) :
\(24 - 12a + 12 + 4b = 0 \implies 36 - 12a + 4b = 0 \)
\(\equiv 9 - 3a + b = 0... (5)\)
Solving (4) and (5), we have :
\(a = 2\frac{1}{2} ; b = -1\frac{1}{2}\)
Coordinates of the centre : \(C(2\frac{1}{2}, -1\frac{1}{2})\).
(b) To get the radius, we can make use of any point on the circle as well as the centre.
Using point \(A(3, 2) & C(2\frac{1}{2}, -1\frac{1}{2})\)
Radius = \(\sqrt{(3 - 2\frac{1}{2})^{2} + (2 + 1\frac{1}{2})^{2}}\)
= \(\sqrt{\frac{1}{4} + \frac{49}{4}}\)
= \(\sqrt{\frac{50}{4}}\)
= \(\frac{5\sqrt{2}}{2} cm\)
(c) Equation of the circle :
\((x - 2\frac{1}{2})^{2} + (y + 1\frac{1}{2})^{2} = (\frac{5\sqrt{2}}{2})^{2}\)
\(x^{2} - 11x + \frac{25}{4} + y^{2} + 3y + \frac{9}{4} = \frac{50}{4}\)
\(x^{2} + y^{2} - 11x + 3y + \frac{34}{4} - \frac{50}{4} = 0\)
\(x^{2} + y^{2} - 11x + 3y - 4 = 0\)
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