(a) Simplify (frac{sqrt{75} – 3}{sqrt{3} + 1}), leaving your answer in the form (a + bsqrt{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.
Explanation
(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} – 3sqrt{3} + 3}{3 – sqrt{3} + sqrt{3} – 1})
= (frac{15 – 5sqrt{3} – 3sqrt{3} + 3}{2})
= (frac{18 – 8sqrt{3}}{2})
= (9 – 4sqrt{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(2f) + c = 0)
(4g + 4f + 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)