A circle with centre (0,0) and radius r has equation x2+y2=r2. A tangent meets the radius at right angles, so its gradient is the negative reciprocal of the radius's gradient.
Read off r:x2+y2=25 has radius 5.
For a tangent at (3,4), the radius has gradient 34.
The tangent's gradient is −43; use the point to find c:y=−43x+425.
Watch out The tangent is perpendicular to the radius: do not use the radius's gradient.
See it worked
Worked exampleNon-calculator·3 marks
A circle has equation x2+y2=49.
(a) Write down the radius of the circle.
(b) Does the point (5,5) lie inside, on or outside the circle? You must show how you get your answer.
(a) x2+y2=r2, so r2=49 and r=7.
(b) Distance from the centre squared: 52+52=50.
50>49, so the point is outside the circle.
Answer (a) 7; (b) Outside: 52+52=50>49.
Your turn
Question 1Non-calculator·3 marks
A circle has its centre at the origin and passes through the point (−6,8).
(a) Write down the equation of the circle.
(b) Find the possible values of y on the circle when x=6.
Question 2Non-calculator·5 marks
A tangent to x2+y2=400 has gradient −3/4 and touches the circle in the first quadrant.
(a) Find its point of contact.
(b) Find the tangent’s y-intercept.
Question 3Non-calculator·3 marks
The line y = 2x + c is tangent to the circle x2 + y2 = 20. Find both possible values of c.
In past papers
42 official questions in 41 papers since 2017. The latest:
