Theory question 109
Surds(a) simplify (√75 − 3)/(√3 + 1), leaving your answers in the form a + b√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; (ii) radius, of the circle.
Show the model answer
Model answer
(a) √75 = 5√3; multiply top and bottom by (√3 − 1):
(5√3 − 3)(√3 − 1)/(3 − 1) = (15 − 5√3 − 3√3 + 3)/2 = (18 − 8√3)/2 = 9 − 4√3
(b) x² + y² + 2gx + 2fy + c = 0 through the points:
14g + 6f + c = −58; 4g + 16f + c = −68; −6g + 6f + c = −18
Solving: g = −2, f = −3, c = −12
(i) x² + y² − 4x − 6y − 12 = 0
(ii) centre (2, 3), r = √(4 + 9 + 12) = 5 units
Explanation
Rationalise with the conjugate; for the circle, substitute each point into the general equation.