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Cambridge IGCSE0606

Coordinate geometry of the circle

Mathematics Additional 0606 Chapter Notes

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Coordinate geometry of the circle
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Coordinate geometry of the circle notes

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In this chapter

  1. 1The equation of a circle
  2. 2Where a line meets a circle
  3. 3Tangent, chord or no intersection
  4. 4Tangents to a circle
  5. 5Intersection of two circles
  6. 6Common chord and an unknown radius

1. The equation of a circle

A circle with centre (a,b) and radius r has the equation below. It is in the List of formulas. Read the centre from the brackets with the signs reversed, and r² from the right-hand side.

A circle may be given as x² + y² + 2gx + 2fy + c = 0. Complete the square to find its centre and radius. If AB is a diameter, the centre is the midpoint of AB and the radius is half the length of AB.

(x-a)² + (y-b)² = r²

x² + y² + 2gx + 2fy + c = 0

centre (-g, -f), r = √(g² + f² - c)

Examiner insight

“Many candidates found the length of the diameter, AB, and then halved it to find the length of the radius. Most of these were successful.” “A few candidates did not correctly recall the equation of the circle. These candidates may have improved if they had checked the List of formulas on page 2 of the examination paper as this formula is given.”

Worked example 14 marks

A circle has equation (x - 3)² + (y - 4)² = 20.

(a) Write down the coordinates of the centre of the circle.

(b) Write down the radius of the circle.

(c) The line AB is a diameter of this circle. The point A has coordinates (-1, 6). Find the coordinates of the point B.

  1. 1

    (a) centre = (3, 4) [1]

  2. 2

    (b) radius = √20 = 2√5 [1]

  3. 3

    (c) the centre is the midpoint of AB: (-1 + x_B)/2 = 3 and (6 + y_B)/2 = 4 [1]

  4. 4

    B = (7, 2) [1]

2. Where a line meets a circle

The points where a line meets a circle satisfy both equations. Substitute the line into the circle to eliminate one unknown. This gives a 3-term quadratic.

Solve it by factorising or with the quadratic formula. Then put each root into the line to find the other coordinate. Pair each x with its own y.

Examiner insight

“Some arithmetic or algebraic slips were made when simplifying work to obtain a quadratic equation in terms of x and k.” “Candidates are recommended to use either a factorisation method or the quadratic formula to solve quadratic equations and then use their calculator as a check.”

Worked example 14 marks

Do not use a calculator. Find the coordinates of the points where the line y = x + 1 meets the circle x² + y² = 25.

  1. 1

    substitute y = x + 1: x² + (x + 1)² = 25 [1]

  2. 2

    2x² + 2x - 24 = 0, so x² + x - 12 = 0 [1]

  3. 3

    (x + 4)(x - 3) = 0, so x = -4 or x = 3 [1]

  4. 4

    y = x + 1 gives the points (-4, -3) and (3, 4) [1]

3. Tangent, chord or no intersection

After substituting, the number of solutions of the quadratic shows how the line meets the circle. Use the discriminant b² - 4ac.

Two solutions: the line meets the circle twice, so it is a chord. One solution: the line is a tangent. No solutions: the line does not intersect the circle.

b² - 4ac > 0 (chord)

b² - 4ac = 0 (tangent)

b² - 4ac < 0 (no intersection)

Examiner insight

“A few candidates obtained an incorrect equation for the circle, with either sign errors or incorrectly equating their equation to 5 rather than 25.” “Some candidates made arithmetic or algebraic slips when using the discriminant of their quadratic equation.”

Worked example 18 marks

A circle has centre (-2, 0) and radius 5. A line has equation y = 2x + k, where k is a constant. The line does not meet the circle. Find the set of possible values of k.

  1. 1

    circle: (x + 2)² + y² = 25 [1]

  2. 2

    substitute: (x + 2)² + (2x + k)² = 25 [1]

  3. 3

    5x² + 4(k + 1)x + k² - 21 = 0 [1]

  4. 4

    no intersection, so b² - 4ac < 0: 16(k + 1)² - 20(k² - 21) < 0 [1]

  5. 5

    -4k² + 32k + 436 < 0, so k² - 8k - 109 > 0 [1]

  6. 6

    solve k² - 8k - 109 = 0: k = (8 ± √(64 + 436))/2 [1]

  7. 7

    k = 4 ± 5√5 [1]

  8. 8

    k < 4 - 5√5 or k > 4 + 5√5 [1]

4. Tangents to a circle

A tangent touches a circle at one point. There are two methods.

Method 1: substitute the line into the circle and set the discriminant equal to zero. Method 2: the tangent is perpendicular to the radius at the point of contact, so its gradient is the negative reciprocal of the gradient of the radius. Then use y - y₁ = m(x - x₁) at that point.

No calculus is expected. Give the equation in the form the question asks for.

m_radius = (y_P - b)/(x_P - a)

m_tangent = -1/m_radius

y - y₁ = m(x - x₁)

Examiner insight

“A few candidates correctly calculated the gradient of the radius but did not take the negative reciprocal when finding the gradient of the tangent.” “Although many correct solutions were seen, many candidates did not appear to read the demand of the question fully and hence did not give the tangent equation in the required form.”

Worked example 18 marks

A circle has centre (3, 1) and radius 1. A line has equation 2y + x = c, where c is a constant.

(a) Given that the line is a tangent to the circle, find the two possible values of c.

(b) When c takes a value between the two values found in part (a), state the relationship between the line and the circle.

  1. 1

    (a) circle: (x - 3)² + (y - 1)² = 1 [1]

  2. 2

    line: x = c - 2y, so (c - 2y - 3)² + (y - 1)² = 1 [1]

  3. 3

    5y² + (10 - 4c)y + (c² - 6c + 9) = 0 [1]

  4. 4

    tangent, so b² - 4ac = 0: (10 - 4c)² - 20(c² - 6c + 9) = 0 [1]

  5. 5

    -4c² + 40c - 80 = 0, so c² - 10c + 20 = 0 [1]

  6. 6

    c = (10 ± √(100 - 80))/2 [1]

  7. 7

    c = 5 ± √5 [1]

  8. 8

    (b) the line meets the circle twice: it is a chord [1]

5. Intersection of two circles

The points where two circles meet satisfy both equations. Subtract one equation from the other. The squared terms cancel, leaving the equation of the common chord, a straight line. Substitute it into one circle to find the points of intersection.

To decide whether two circles intersect, touch or do not intersect, find the distance d between the centres. Compare it with the sum of the radii.

d = √((x₂ - x₁)² + (y₂ - y₁)²)

d > r₁ + r₂ (no intersection)

d = r₁ + r₂ (touch)

|r₁ - r₂| < d < r₁ + r₂ (intersect)

Examiner insight

“Some candidates subtracted the equations of the circles to find the correct equation of the common chord. This was an efficient approach.” “Solutions which contained a diagram made it easier for candidates to consider the distance between the centres of the circles and the sum of the radii.”

Worked example 18 marks

Find the coordinates of the points of intersection of the circles with equations x² + 2x + y² - 4y - 20 = 0 and x² - 12x + y² - 6y + 20 = 0.

  1. 1

    subtract the equations to eliminate x² and y² [1]

  2. 2

    common chord: 14x + 2y - 40 = 0, so y = 20 - 7x [1]

  3. 3

    substitute into the first circle: x² + 2x + (20 - 7x)² - 4(20 - 7x) - 20 = 0 [1]

  4. 4

    x² + 2x + 400 - 280x + 49x² - 80 + 28x - 20 = 0 [1]

  5. 5

    50x² - 250x + 300 = 0, so x² - 5x + 6 = 0 [1]

  6. 6

    (x - 2)(x - 3) = 0 [1]

  7. 7

    x = 2 gives y = 6; x = 3 gives y = -1 [1]

  8. 8

    the points are (2, 6) and (3, -1) [1]

6. Common chord and an unknown radius

The common chord is the line through both points of intersection. Its equation may be given. Substitute it into the circle whose equation you know, to eliminate one unknown and find a point of intersection.

That point lies on the other circle too. Its distance from the other centre is that circle's radius. Write r² on the right-hand side, not r.

r² = (x_P - a)² + (y_P - b)²

Examiner insight

“Some candidates who subtracted the equations of the circles to find the correct equation of the common chord could improve by setting out each step clearly. For example, the relationship y = –7x was stated by candidates who equated the two equations and incorrectly cancelled the –20 on one side with the +20 on the other.”

Worked example 19 marks

Solutions to this question by accurate drawing will not be accepted.

A circle C₁ has centre (3, 5). The point P(1, 4) lies on circle C₁.

A circle C₂ has centre (12, 8). The line with equation y = 19 - 3x is the common chord of circles C₁ and C₂.

Find the radius of circle C₂.

  1. 1

    r² = (1 - 3)² + (4 - 5)² = 5, so C₁: (x - 3)² + (y - 5)² = 5 [1]

  2. 2

    substitute y = 19 - 3x to eliminate y: (x - 3)² + (19 - 3x - 5)² = 5 [1]

  3. 3

    x² - 6x + 9 + 196 - 84x + 9x² = 5 [1]

  4. 4

    10x² - 90x + 200 = 0 [1]

  5. 5

    x² - 9x + 20 = 0, so (x - 4)(x - 5) = 0 [1]

  6. 6

    x = 4 or x = 5 [1]

  7. 7

    y = 19 - 3x gives the points (4, 7) and (5, 4) [1]

  8. 8

    (4, 7) is on C₂: r² = (4 - 12)² + (7 - 8)² [1]

  9. 9

    r² = 65, so r = √65 [1]

Recap

  • Circle: (x-a)² + (y-b)² = r² has centre (a, b) and radius r. Complete the square if it is not in this form.
  • A diameter has the centre as its midpoint. The radius is half the diameter.
  • Line and circle: substitute, get a 3-term quadratic, solve, then find the other coordinate.
  • Discriminant: >0 chord, =0 tangent, <0 no intersection.
  • Tangent: gradient is the negative reciprocal of the gradient of the radius, or discriminant = 0.
  • Two circles: subtract the equations to get the common chord, then substitute.
  • Compare the distance between the centres with the sum of the radii to decide if circles touch or intersect.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write down the centre and the radius of the circle (x + 2)² + (y - 5)² = 49.2 marks
  2. Do not use a calculator. A circle has centre (1, -3) and passes through the point (4, 1). Find its equation.3 marks
  3. Find the centre and the radius of the circle x² + y² - 6x + 4y - 12 = 0.3 marks
  4. Find the coordinates of the points where the line y = x + 2 meets the circle x² + y² = 10.4 marks
  5. Determine whether the line y = x + 6 is a tangent to the circle x² + y² = 16, a chord of it, or neither.4 marks
  6. Do not use a calculator. The point (5, 3) lies on the circle (x - 2)² + (y + 1)² = 25. Find the equation of the tangent to the circle at this point. Give your answer in the form ax + by = c.4 marks
  7. Do not use a calculator. The line y = 2x + k is a tangent to the circle x² + y² = 5. Find the two possible values of k.5 marks
  8. Circle A has centre (0, 0) and radius 3. Circle B has centre (6, 8) and radius 7. Decide whether the circles intersect, touch or do not intersect.3 marks
  9. Do not use a calculator. A garden plan uses coordinates in metres. Two sprinklers each water a circular region. The points A(-1, 1) and B(7, 7) are the ends of a diameter of the first region. The second sprinkler is at the point (9, 7) and its region has radius √10 m. (a) Find the equation of the edge of the first region. (b) Find the coordinates of the two points where the edges of the regions cross. (c) Find the exact length of the straight line joining these two points.10 marks
  10. Do not use a calculator. A plan of a park uses coordinates in metres. A circular pond has its centre at the origin O and radius 2√5 m. A lamp stands at the point P(0, 10). Two straight beams of light from P each touch the edge of the pond at one point. A beam has equation y = mx + 10, where m is a constant. (a) Find the two possible values of m. (b) Find the coordinates of the point where each beam touches the edge of the pond. (c) Find the exact length of each beam from P to the edge of the pond.10 marks

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