UK K12 GCSE/A-Level: Year 12 A-Level Lower Sixth Mathematics - Pure Coordinate Geometry Circles

Learning Objectives

By the end of this topic, students will be able to: - Define and identify the key properties of a circle in the Cartesian plane. - Use the equation of a circle in standard form to find the centre and radius. - Apply the equation of a circle to solve problems involving tangents, chords, and intersections with lines. - Prove and apply theorems related to circles, including the power of a point and the intersecting chord theorem. - Analyze and interpret geometric relationships in the context of real-world problems.

Core Concepts

A circle is a set of points in a plane that are equidistant from a fixed point, known as the centre. The radius of a circle is the distance from the centre to any point on the circle. The equation of a circle in standard form is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the centre of the circle and r is the radius.

The equation of a circle can be derived from the distance formula:

d^2 = (x - h)^2 + (y - k)^2

where d is the distance from the centre to a point (x, y) on the circle.

The circle can be rotated by ?/2 radians (90°) about its centre, resulting in a new circle with the same radius but a different orientation.

The power of a point theorem states that if two chords intersect within a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Worked Examples

Example 1: Finding the Centre and Radius of a Circle

Find the centre and radius of the circle with equation:

x^2 + y^2 - 4x + 6y - 12 = 0

Rearrange the equation to standard form:

(x - 2)^2 + (y + 3)^2 = 25

The centre of the circle is (2, -3) and the radius is ?25 = 5.

Example 2: Finding the Equation of a Circle

Find the equation of the circle with centre (1, 2) and radius 3.

Using the equation of a circle in standard form:

(x - 1)^2 + (y - 2)^2 = 3^2

Simplifying:

x^2 - 2x + 1 + y^2 - 4y + 4 = 9

Rearranging:

x^2 + y^2 - 2x - 4y - 4 = 0

Example 3: Applying the Power of a Point Theorem

Two chords, AB and CD, intersect within a circle at point E. If AE = 4, EB = 6, CE = 3, and ED = 8, find the length of CD.

Using the power of a point theorem:

AE × EB = CE × ED

Substituting the given values:

4 × 6 = 3 × 8

24 = 24

Since the equation is true, the lengths of the segments of the chords are consistent with the power of a point theorem.

Common Misconceptions

• Students may confuse the equation of a circle with the equation of a parabola or ellipse.
• Students may forget to square the radius when rearranging the equation of a circle to standard form.
• Students may misapply the power of a point theorem, forgetting that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Exam Tips

• Always check the units of the answer when solving problems involving circles.
• Use the equation of a circle to find the centre and radius, and vice versa.
• Apply the power of a point theorem carefully, ensuring that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
• Use geometric relationships to solve problems involving circles, such as the intersecting chord theorem.

MCQs

MCQ 1 [F]

What is the equation of a circle with centre (2, 3) and radius 4? A) (x - 2)^2 + (y - 3)^2 = 4 B) (x - 2)^2 + (y - 3)^2 = 16 C) (x + 2)^2 + (y + 3)^2 = 4 D) (x + 2)^2 + (y + 3)^2 = 16

Correct answer: B) (x - 2)^2 + (y - 3)^2 = 16

Why the distractors fail: A) The equation is missing the squared radius. C) The equation has the wrong sign for the x and y terms. D) The equation has the wrong sign for the squared radius.

MCQ 2 [H]

Two chords, AB and CD, intersect within a circle at point E. If AE = 4, EB = 6, CE = 3, and ED = 8, find the length of CD. A) 6 B) 8 C) 12 D) 16

Correct answer: C) 12

Why the distractors fail: A) The product of the lengths of the segments of one chord does not equal the product of the lengths of the segments of the other chord. B) The product of the lengths of the segments of one chord does not equal the product of the lengths of the segments of the other chord. D) The product of the lengths of the segments of one chord does not equal the product of the lengths of the segments of the other chord.

MCQ 3 [F]

What is the power of a point theorem? A) If two chords intersect within a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. B) If two chords intersect within a circle, the sum of the lengths of the segments of one chord equals the sum of the lengths of the segments of the other chord. C) If two chords intersect within a circle, the difference of the lengths of the segments of one chord equals the difference of the lengths of the segments of the other chord. D) If two chords intersect within a circle, the ratio of the lengths of the segments of one chord equals the ratio of the lengths of the segments of the other chord.

Correct answer: A) If two chords intersect within a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Why the distractors fail: B) The theorem states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord, not the sum. C) The theorem states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord, not the difference. D) The theorem states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord, not the ratio.

MCQ 4 [H]

What is the equation of a circle with centre (1, 2) and radius 3? A) (x - 1)^2 + (y - 2)^2 = 3 B) (x - 1)^2 + (y - 2)^2 = 9 C) (x + 1)^2 + (y + 2)^2 = 3 D) (x + 1)^2 + (y + 2)^2 = 9

Correct answer: B) (x - 1)^2 + (y - 2)^2 = 9

Why the distractors fail: A) The equation is missing the squared radius. C) The equation has the wrong sign for the x and y terms. D) The equation has the wrong sign for the squared radius.

MCQ 5 [F]

What is the distance from the centre of a circle to a point on the circle? A) Radius B) Diameter C) Circumference D) Chord

Correct answer: A) Radius

Why the distractors fail: B) The diameter is twice the radius. C) The circumference is the distance around the circle. D) A chord is a line segment connecting two points on the circle.

Short-answer questions

Question 1

Find the equation of the circle with centre (3, 4) and radius 2.

Question 2

Two chords, AB and CD, intersect within a circle at point E. If AE = 4, EB = 6, CE = 3, and ED = 8, find the length of CD.

Question 3

What is the power of a point theorem?

Question 4

Find the distance from the centre of a circle to a point on the circle.

Question 5

What is the equation of a circle with centre (1, 2) and radius 3?