Ch 2 · 8 Coordinate Geometry of the Circle · O Level Additional Mathematics 4037
Topic 2 of 2 in Coordinate Geometry — O Level Additional Mathematics 4037 — 56 lessons.
Introduction
Two mobile-phone masts each cover a circular area. Where do the coverage areas overlap? Does a straight road pass through an area, just touch it, or miss it? Questions like these are about circles in coordinates, and they combine everything from straight lines and quadratics. (Syllabus topic 8.)
Key ideas — The equation of a circle
(x - a)^2 + (y - b)^2 = r^2 centre (a, b),\ radius r
🧑🏫 Teacher's explanation
Imagine you are standing at a fixed point (a, b) and you spin a rope of length r around you. Every point (x, y) that the end of the rope touches forms a circle. The equation (x - a)^2 + (y - b)^2 = r^2 is simply the Pythagorean theorem in disguise: it says that for any point on the circle, the horizontal distance squared plus the vertical distance squared must equal the radius squared.
📖 Definitions
🔍 Worked example — Easy
Find the centre and radius of the circle with equation (x - 3)^2 + (y + 4)^2 = 49.
🔍 Worked example — Standard
Find the centre and radius of the circle x^2 + y^2 - 6x + 10y + 9 = 0.
🔍 Worked example — Exam-style
A circle has the equation x^2 + y^2 + 4x - 2y - 11 = 0. Determine if the point (1, 2) lies inside, on, or outside the circle.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — A line and a circle
Substitute the line into the circle to get a quadratic in one variable. Its discriminant decides:
🧑🏫 Teacher's explanation
Imagine a circle drawn on a piece of paper and a straight ruler placed over it. Depending on where you slide the ruler, it might cross the circle at two points (like a chord), just barely touch the edge at one single point (like a tangent), or miss the circle entirely.
📖 Definitions
🔍 Worked example — Easy
Determine the number of points of intersection between the circle x^2 + y^2 = 25 and the line y = x + 1.
🔍 Worked example — Standard
Show that the line y = 5 is a tangent to the circle x^2 + y^2 - 4x - 6y + 9 = 0.
🔍 Worked example — Exam-style
Find the equation of the tangent to the circle x^2 + y^2 = 10 at the point (1, 3).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Find the coordinates of the center and the radius of the circle. [3] (b) Show that the line x - y + 1 = 0 is a tangent to the circle. [3] [6 marks]
Key ideas — Tangents
The tangent at a point P on the circle is perpendicular to the radius CP. So: find the gradient of CP, take the negative reciprocal, and use y - y_1 = m(x - x_1). No calculus is needed. To find tangents with a given gradient, use the discriminant = 0 condition.
🧑🏫 Teacher's explanation
Imagine you are spinning a ball on a string. If you suddenly let go of the string, the ball doesn't fly straight away from you; it flies off in a straight line that just barely 'grazes' the circular path you were making. That straight line is the tangent.
📖 Definitions
🔍 Worked example — Easy
A circle has center C(3, -2) and a point P(7, 1) lies on the circle. Find the equation of the tangent to the circle at point P.
🔍 Worked example — Standard
Find the equation of the tangent to the circle x^2 + y^2 = 25 that has a gradient of 2.
🔍 Worked example — Exam-style
A circle has center (0,0) and radius 5. Find the equation of the tangent at the point where the circle intersects the line y = x.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Two circles
Let d be the distance between the centres and r_1, r_2 the radii:
🧑🏫 Teacher's explanation
Imagine two soap bubbles floating towards each other. If they are far apart, they don't touch. If they just barely graze each other, they touch at a single point. If they overlap, they share a curved boundary, but the 'seam' where they meet is actually a perfectly straight line called the common chord.
📖 Definitions
🔍 Worked example — Easy
Circle C_1 has center (0,0) and radius 3. Circle C_2 has center (5,0) and radius 2. Determine if they intersect, touch, or do not meet.
🔍 Worked example — Standard
Find the equation of the common chord for the circles x^2 + y^2 = 25 and x^2 + y^2 - 4x - 4y + 4 = 0.
🔍 Worked example — Exam-style
Two circles have centers A(1, 2) and B(4, 6). Their radii are r_1 = 3 and r_2 = 2. Determine the number of intersection points.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Circle 1: x^2 + y^2 - 2x - 4y - 20 = 0 Circle 2: x^2 + y^2 + 4x + 2y - 20 = 0 (a) Find the equation of the common chord of the two circles. (b) Determine whether the circles intersect, touch, or do not meet. [5 marks]
Worked examples — Example 1: a chord and a tangent
Circle x^2 + y^2 - 4x + 2y - 20 = 0 (centre (2, -1), radius 5).
🔍 Worked example — Easy
A circle has the equation x^2 + y^2 - 4x + 2y - 20 = 0. Determine if the line y = x + 2 is a chord, a tangent, or does not intersect the circle.
🔍 Worked example — Standard
Find the equation of the tangent to the circle x^2 + y^2 - 4x + 2y - 20 = 0 at the point (5, 3).
🔍 Worked example — Exam-style
Show that the line x - 2y + 10 = 0 is a tangent to the circle x^2 + y^2 = 25.
⚠️ Common errors
Worked examples — Example 2: tangents with a given gradient
Find k so that y = 2x + k is a tangent to x^2 + y^2 = 5. x^2 + (2x + k)^2 = 5 ⇒ 5x^2 + 4kx + k^2 - 5 = 0. Tangent: 16k^2 - 20(k^2 - 5) = 0 ⇒ k^2 = 25, so k = ± 5.
🔍 Worked example — Exam-style
A circle is centered at the origin (0,0) with radius √(10). Find the equations of the tangents to the circle that have a gradient of 3.
⚠️ Common errors
Worked examples — Example 3: two circles and their common chord
C_1: x^2 + y^2 = 25 (centre O, radius 5). C_2: x^2 + y^2 - 12x - 6y + 25 = 0, i.e. (x - 6)^2 + (y - 3)^2 = 20 (centre (6, 3), radius 2sqrt5 ≈ 4.47).
🔍 Worked example — Easy
Circle C_1 has equation x^2 + y^2 = 25. Circle C_2 has equation x^2 + y^2 - 12x - 6y + 25 = 0. Find the equation of the common chord.
🔍 Worked example — Standard
Determine if C_1: x^2 + y^2 = 25 and C_2: (x - 6)^2 + (y - 3)^2 = 20 intersect at two points, one point, or no points.
🔍 Worked example — Exam-style
Find the coordinates of the intersection points of x^2 + y^2 = 25 and x^2 + y^2 - 12x - 6y + 25 = 0.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (x + 3)^2 + (y - 4)^2 = 25: centre (-3, 4), radius 5. (2) centre (4, 6), radius 5: (x - 4)^2 + (y - 6)^2 = 25. (3) y = 10 - x: (x - 1)^2 + (8 - x)^2 = 9 ⇒ 2x^2 - 18x + 56 = 0, discriminant 324 - 448 < 0. (4) d = 3 = 2 + 1 = r_1 + r_2, so they touch externally, at (2, 0). (5) radius gradient tfrac13, tangent gradient -3: y + 1 = -3(x - 4), i.e. 3x + y = 11.