Solving a linear equation with a quadratic or a circle by substitution, interpreting the solutions as intersection points, and the condition for a tangent.
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.
Where does a line meet a circle? Substitute, solve a quadratic, and you have the coordinates — and the discriminant tells you whether it meets at all.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Move the line and count the solutions.
Copy these into your notebook in your own words — that is how they stick.
Linear → substitute → quadratic → solve → back-substitute in the LINEAR equation.
Give pairs (x, y). Two solutions mean two intersection points.
D > 0 cuts at two points; D = 0 tangent; D < 0 misses.
(x − a)² + (y − b)² = r²; substitution gives a quadratic.
Use the linear equation (not the quadratic) to find the second coordinate.
Substitute each pair into the non-linear equation.
Define two variables, form one linear and one non-linear equation.
For a tangent, the repeated root gives the x-coordinate of the point of contact.
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 (25 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [15 marks]
Answer all parts.
(a) Solve the simultaneous equations y = 2x + 1 and y = x² − 2. [5]
(b) The line y = x − 1 meets the circle x² + y² = 25 at A and B. Find the coordinates of A and B and the length of AB. [7]
(c) The mid-point of AB is M. Show that OM is perpendicular to AB where O is the origin. [3]
Question 2 [10 marks]
The line y = x + k and the curve y = x² + 3x + 5 are given.
(a) Show that the x-coordinates of the points of intersection satisfy x² + 2x + (5 − k) = 0. [2]
(b) Find the value of k for which the line is a tangent to the curve. [3]
(c) Find the coordinates of the point of contact. [3]
(d) State the range of values of k for which the line does not meet the curve. [2]
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.