Writing and reading the equation of a circle, finding the centre and radius, tangents and normals at a point, and where a line meets a circle.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
A circle is the set of all points a fixed distance from a centre. Turn that sentence into an equation and tangents, normals and intersections become routine algebra.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Move a point round the circle.
Copy these into your notebook in your own words — that is how they stick.
(x − a)² + (y − b)² = r².
x² + y² + 2fx + 2gy + c = 0; centre (−f, −g); r = √(f² + g² − c).
Substitute the point: it satisfies the equation.
Perpendicular to the radius at the point of contact: m_t = −1/m_radius.
Passes through the point and the centre.
Substitute; discriminant: > 0 cuts, = 0 tangent, < 0 misses.
The perpendicular from the centre to a chord bisects it.
Test the point in the circle equation and the tangent equation.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (27 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [14 marks]
A circle has equation x² + y² − 4x + 6y − 12 = 0.
(a) Find the coordinates of the centre and the radius. [3]
(b) Show that the point A(5, 1) lies on the circle. [2]
(c) Find the equation of the tangent at A. [4]
(d) Find the equation of the normal at A. [2]
(e) The line y = 1 meets the circle again at B. Find the coordinates of B. [3]
Question 2 [13 marks]
The line y = x + 2 meets the circle x² + y² = 20 at P and Q.
(a) Find the coordinates of P and Q. [5]
(b) Calculate the length of the chord PQ and the coordinates of its midpoint M. [3]
(c) Show that OM is perpendicular to PQ, where O is the origin. [2]
(d) Find the values of c for which y = 2x + c is a tangent to the circle. [3]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.