(a) Simplify \(\frac{\sqrt{75} - 3}{\sqrt{3} + 1}\), leaving your answer in the form \(a + b\sqrt{c}\); where a, b and c are rational numbers.
(b) The points (7, 3), (2, 8) and (-3, 3) lie on a circle. Find the (i) equation and (ii) radius of the circle.
\(\frac{\sqrt{75} - 3}{\sqrt{3} + 1}\)
Rationalizing, \((\frac{\sqrt{75} - 3}{\sqrt{3} + 1})(\frac{\sqrt{3} - 1}{\sqrt{3} - 1}\)
\(\frac{\sqrt{3 \times 75} - \sqrt{75} - 3\sqrt{3} + 3}{3 - \sqrt{3} + \sqrt{3} - 1}\)
= \(\frac{15 - 5\sqrt{3} - 3\sqrt{3} + 3}{2}\)
= \(\frac{18 - 8\sqrt{3}}{2}\)
= \(9 - 4\sqrt{3}\)
(b)(i) The general form of the circle equation is \(x^{2} + y^{2} + 2gx + 2fy + c = 0\) where (x, y) is a point on the circle.
\(\therefore A(7, 3) = 7^{2} + 3^{2} + 7(2g) + 3(2f) + c = 0\)
\(14g + 6f + c = -58 .... (1)\)
\(B(2, 8) = 2^{2} + 8^{2} + 2(2g) + 2(8f) + c = 0\)
\(4g + 16f + c = -68 ..... (2)\)
\(C(-3, 3) = -3^{2} + 3^{2} - 3(2g) + 3(2f) + c = 0\)
\(6f - 6g + c = -18 ....(3)\)
Solving the equations, we get \(g = -2; f = -3; c = -12\)
\(\text{The equation of the circle is} x^{2} + y^{2} - 4x - 6y - 12 = 0\).
(b)(ii) Radius = \(\sqrt{g^{2} + f^{2} - c}\)
= \(\sqrt{-2^{2} + (-3^{2}) - (-12)}\)
\(\sqrt{25} = 5 units\)
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