Equation of a circle GCSE lesson: x² + y² = r² and the tangent at a point

The equation of a circle: practice

Show your working on paper. Write fractions with /, such as -1/2, and negative numbers with a minus sign. The stretch is optional.

Diagnostic question: choose and explain

1. The point (2, 4) is on the circle x² + y² = 20. What is the gradient of the tangent there?
   1. −1/2
   2. 2
   3. −2
   4. 1/2

Practice questions

1. x² + y² = 49 is a circle with radius ___
2. The circle with centre (0, 0) and radius 6 is x² + y² = ___
3. 2x² + 2y² = 50 is a circle. What is its radius?
4. The point (−2, 5) is on the circle x² + y² = k. What is k?
5. Where is the point (3, −3) compared with the circle x² + y² = 16?
   1. Inside
   2. On it
   3. Outside
6. Which point is on the circle x² + y² = 100?
   1. (6, −8)
   2. (4, 6)
   3. (10, 10)
7. The tangent to x² + y² = 10 at (3, 1) is y = ___x + ___
8. The tangent to x² + y² = 20 at (4, 2) crosses the y-axis at (0, c). What is c?
9. The tangent to x² + y² = 25 at (0, 5) is the line y = ___
10. Stretch: the line y = x + 1 crosses x² + y² = 25 at the smaller x = ___ and the larger x = ___

Support: For question 3: divide every term by 2 first, so it reads x² + y² = … . The number on the right is r²: take its square root for the radius.

The equation of a circle: exit

Show your working; one sentence for the second part.

1. Find the equation of the tangent to x² + y² = 17 at the point (1, 4). Then say in one sentence why the tangent has the gradient you used.

Exit check

One tangent and one sentence show who links the tangent's gradient to the radius.

Find the equation of the tangent to x² + y² = 17 at the point (1, 4).

The equation of a circle: points and tangents

Show your working. For each point, work out x² + y² and compare it with r². For each tangent, write the radius's gradient first.