Equation of a tangent to a circle

The Maths Habit

Each circle has its centre at (0, 0).

  1. The radius is 9. Write the equation.

    x2 + y2 = 81

  2. Write down the radius of x2 + y2 = 49

    7

  3. Is (5, −2) inside, on or outside x2 + y2 = 29?

    On: 25 + 4 = 29

A bike wheel rolls along flat ground. How many points touch the ground at once? What angle does the spoke to that point make with the ground?

One point

90°: the spoke is straight up and down

Flip it and change the sign

  1. 12

    −2

  2. 3

    −13

  3. −25

    52

A tangent meets the radius at 90°

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

gradient of the radius14
gradient of the tangent−4
put the point into y = mx + cc =17
Answery = −4x + 17

Find the tangent to x2 + y2 = 45 at (6, 3).

gradient of the radius12
gradient of the tangent−2
put the point into y = mx + cc =15
Answery = −2x + 15

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

gradient of the radius14
gradient of the tangent−4
put the point into y = mx + cc =17
Answery = −4x + 17

Find the tangent to x2 + y2 = 52 at (6, 4).

gradient of the radius23
gradient of the tangent−32
put the point into y = mx + cc =13
Answery = −32x + 13

Equation of a tangent to a circle worksheet, page 1

Answer sheet

  1. The tangent to x2 + y2 = 10 at (3, 1) crosses the x axis at P. Find P.

    (103, 0)

  2. Is the point (3, 3) inside, on or outside the circle x2 + y2 = 20?

    Inside: 18 is less than 20

  1. P(3, −1) is on the circle x2 + y2 = 10. Find the gradient of the tangent at P.

    3

  2. Find the equation of the tangent to x2 + y2 = 10 at P(3, −1).

    y = 3x − 10

  3. Fin says the tangent to x2 + y2 = 20 at (2, 4) has a gradient of 2. Is Fin right? Explain.

    No: the tangent is −12
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