AS Level Mathematics · Pure Mathematics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
This topic links algebra to straight lines and circles. A line is fixed by one point and its gradient, so most line questions come down to finding those two things. Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1.
A circle is fixed by its centre and radius. If the equation is given in expanded form, complete the square in x and in y to read them off. Two circle facts are used often: a tangent is perpendicular to the radius at the point of contact, and the angle in a semicircle is 90°.
Points where two graphs meet are the solutions of their equations solved together. The discriminant tells you how many such points there are.
Students read the centre of (x + 4)2 + (y − 1)2 = 25 as (4, −1) with radius 25. The signs change, and the right-hand side is r2: the centre is (−4, 1) and the radius is 5.
Answer: 5x + 2y = 22
Try each one, then open the answer.
Answer: B. x2 + x + 4 = 3x + c gives x2 − 2x + (4 − c) = 0. No intersection means 4 − 4(4 − c) < 0, so 1 − 4 + c < 0 and c < 3. At c = 3 the line is a tangent and does meet the curve, so the inequality is strict.
Answer: A. Midpoint (2, 4); gradient of AB = 4/6 = 2/3, so the perpendicular gradient is −3/2. y − 4 = −3/2 (x − 2) gives 2y − 8 = −3x + 6, i.e. 3x + 2y = 14.
Answer: C. AB = √((4 − (−2))2 + (−5 − 3)2) = √(36 + 64) = √100 = 10. Subtract coordinates carefully with the negative signs.
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