These twelve questions are original and run from easy to harder. They cover reading a circle, completing the square, intersections, tangents, tangency conditions and building equations. Attempt each on paper first, then open the worked answer.
Draw a quick sketch for every question. Check the notation and calculator rules for your exam year on the Cambridge syllabus page. Start from the module overview if you need the study order.
Warm-up: reading and completing
1. Write down the centre and radius of (x − 3)² + (y + 5)² = 49.
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The centre has x = 3 and y = −5, and r² = 49.
Centre (3, −5), radius 7.
2. Find the centre and radius of x² + y² − 10x + 4y + 13 = 0.
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(x² − 10x + 25) + (y² + 4y + 4) = −13 + 25 + 4, so (x − 5)² + (y + 2)² = 16.
Centre (5, −2), radius 4.
3. Find the centre and radius of 3x² + 3y² + 6x − 12y − 12 = 0.
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Divide by 3: x² + y² + 2x − 4y − 4 = 0. Then (x + 1)² + (y − 2)² = 4 + 1 + 4 = 9.
Centre (−1, 2), radius 3.
4. Explain why x² + y² − 6x + 2y + 15 = 0 is not the equation of a circle.
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(x − 3)² + (y + 1)² = −15 + 9 + 1 = −5. A sum of two squares cannot be negative, so no circle exists.
Core: intersections and tangents
5. Find the points where the line y = x + 2 meets the circle x² + y² = 20.
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x² + (x + 2)² = 20 gives 2x² + 4x − 16 = 0, so x² + 2x − 8 = 0 and (x + 4)(x − 2) = 0. Then x = 2 or x = −4.
From y = x + 2: y = 4 or y = −2.
Points (2, 4) and (−4, −2). Check: 4 + 16 = 20 ✓ and 16 + 4 = 20 ✓.
6. The line y = 2x + 8 and the circle x² + y² = 16 are drawn. Use the discriminant to show the line cuts the circle twice, then find the points.
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x² + (2x + 8)² = 16 gives 5x² + 32x + 48 = 0.
Discriminant: 32² − 4 × 5 × 48 = 1024 − 960 = 64, which is positive, so there are two points.
x = (−32 ± 8) ÷ 10, so x = −2.4 or x = −4. Then y = 2(−2.4) + 8 = 3.2 and y = 2(−4) + 8 = 0.
Points (−2.4, 3.2) and (−4, 0). Check: 5.76 + 10.24 = 16 ✓ and 16 + 0 = 16 ✓.
7. Find the equation of the tangent to (x − 1)² + (y + 1)² = 13 at the point (3, 2).
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Point check: 2² + 3² = 13 ✓. Radius gradient from (1, −1) to (3, 2) is 3/2, so tangent gradient is −2/3.
y − 2 = −2/3 (x − 3), so 3y − 6 = −2x + 6.
2x + 3y = 12. Check: 6 + 6 = 12. ✓
8. The circle x² + y² − 6x − 8y = 0 passes through the origin. Find the equation of the tangent at the origin.
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Completing the square: (x − 3)² + (y − 4)² = 25, so the centre is (3, 4). The radius gradient from the centre to (0, 0) is 4/3, so the tangent gradient is −3/4.
The line passes through the origin, so y = −3/4 x, which is 3x + 4y = 0.
Check: the distance from (3, 4) to the line is |9 + 16| ÷ 5 = 5, equal to the radius. ✓
Stretch: parameters and construction
9. Find the values of k for which y = 3x + k is a tangent to x² + y² = 10.
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x² + (3x + k)² = 10 gives 10x² + 6kx + k² − 10 = 0.
Discriminant: 36k² − 40(k² − 10) = 400 − 4k². Set to 0: k² = 100.
k = 10 or k = −10. Check: |k| ÷ √10 = √10 gives |k| = 10. ✓
10. The line y = 2x + k and the circle (x − 1)² + y² = 5 are given. Find the values of k for which the line is a tangent, and state the range of k for which it cuts the circle twice.
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(x − 1)² + (2x + k)² = 5 gives 5x² + (4k − 2)x + k² − 4 = 0.
Discriminant: (4k − 2)² − 20(k² − 4) = 16k² − 16k + 4 − 20k² + 80 = −4k² − 16k + 84.
Set to 0: k² + 4k − 21 = 0, so (k + 7)(k − 3) = 0 and k = 3 or k = −7.
For two points the discriminant is positive, so k² + 4k − 21 < 0, which gives −7 < k < 3.
Check k = 0: discriminant 84 > 0 ✓.
11. Write down the equation of the circle with diameter PQ, where P is (−3, 2) and Q is (5, −4).
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Centre (midpoint): (1, −1). PQ² = 8² + 6² = 100, so PQ = 10 and r = 5.
(x − 1)² + (y + 1)² = 25. Check: P gives 16 + 9 = 25 ✓ and Q gives 16 + 9 = 25 ✓.
12. A circle passes through O(0, 0), A(8, 0) and B(0, 6). Find its equation, its centre and its radius, and explain why AB is a diameter.
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Let the equation be x² + y² + Dx + Ey + F = 0. Since O is on the circle, F = 0.
A: 64 + 8D = 0, so D = −8. B: 36 + 6E = 0, so E = −6.
x² + y² − 8x − 6y = 0 becomes (x − 4)² + (y − 3)² = 25. Centre (4, 3), radius 5.
AB has length √(64 + 36) = 10, which equals twice the radius. The midpoint of AB is (4, 3), the centre, so AB is a diameter. This agrees with the right angle at O.
If you got these wrong
| What went wrong | Go back to |
|---|---|
| Wrong signs in the centre, or r² quoted as r (questions 1 to 4) | Recover centre and radius by completing the square |
| Lost a term when expanding a bracket, or wrong second coordinate (5, 6) | Find intersections of a line and circle |
| Used the radius gradient as the tangent gradient (7, 8) | Use a tangent perpendicular to the radius |
| Stopped at one value of k, or used greater than zero (9, 10) | Solve a parameter condition for tangency |
| Used the diameter as the radius, or could not find the centre (11, 12) | Translate a geometric condition into a circle equation |
Use the line and circle intersection explorer to see each intersection case, and the non-calculator working trainer for the exact arithmetic. Record each slip in the mistake log and retest queue and retry a similar question after a few days.
If questions 9 to 12 are where your progress slows, the gap is usually choosing a plan rather than doing the algebra. A teacher can practise that with you in online one-to-one Additional Mathematics tuition.