Chapter 4 · Geometry and Equations · Concept 3
4-18 Equation of a Circle (3)
Now the clue is that the circle just touches something — an axis — or that its centre is stuck on a line. Both clues say the same kind of thing: they tie the centre down, and once the centre is nearly known, everything else follows.
Guiding question
A coin resting in the corner of a table. Where is its middle?
Learning outcome. Find the equation of a circle that is tangent to an axis, or whose centre lies on a given line.
Touching means the radius reaches exactly that far
tangent to the x-axis
r = |b|
the up-down coordinate of the centre
tangent to the y-axis
r = |a|
the left-right coordinate of the centre
It crosses over, and that is the trap.
The x-axis question is answered by the y coordinate. Say it out loud once: to reach down to the x-axis, a circle must reach as far as its own height. Centre (4, −2) is 2 units below the x-axis, so touching it means r = 2.
Tangent to BOTH axes.
Then |a| = |b| = r, so the centre sits on a diagonal: (r, r) or (−r, r) or (−r, −r) or (r, −r). Which corner is decided by the point you are given — the circle cannot cross an axis it is only touching, so the point and the centre live in the same quarter of the plane.
Centre stuck on a line.
If the centre is on y = x + 5, write it as (a, a + 5) — one unknown instead of two. Then use the points on the circle to pin a down. The same idea as the corner: every clue about the centre removes a letter.
Every “touches both axes and passes through a point” question ends in a quadratic, and a quadratic has two roots. Both draw a real circle.
A circle must touch both axes and pass through this point. Its centre is therefore in the top left corner, the same distance from each axis.
2. Find the circles through (−2, 1) that are tangent to both the x-axis and the y-axis.
- 1
Decide which corner the centre is in.
(−2, 1) is left of the y-axis and above the x-axisA circle only touching the axes never crosses them, so it stays in one quarter — the same quarter as the point. Here that is x negative, y positive.
Work with a partner. For question 3, one of you writes the centre as (a, a + 5) and the other sketches the line y = x + 5 with the two points on it. Compare where you each think the centre must sit before either of you does any algebra.
1–6. Tangent to one axis.
Notice 2 and 3 give the same radius from different centres.
(3, −4) and (5, −4) are both 4 below the x-axis, so both need r = 4. Sliding a circle sideways does not change how far it has to reach downwards. That is the clearest possible proof that only one coordinate matters.
7–9. Tangent to both axes — two answers each.
10–11. Centre on a line.
An arc-based radiotherapy gantry rotates in a circular path. Its centre of rotation is constrained to the track along the x-axis, and the trajectory must pass through two spatial calibration checkpoints, P1(−2, 2) and P2(1, 1). Find the circle’s standard equation, its centre of rotation, and the exact beam radius.
Using the wrong coordinate for the axis.
Tangent to the x-axis uses the y coordinate. It crosses over, and it costs marks every year.
Giving only one of the two circles.
Every "tangent to both axes" question here ends in a quadratic with two positive roots. Both are real circles and both are needed.
Putting the centre in the wrong corner.
The centre must be in the same quarter of the plane as the given point. Through (−1, −2) the centre is (−r, −r), so BOTH coordinates are negative.
Keeping two unknowns when the question gave you a line.
"Centre on y = 2x − 6" means the centre is (a, 2a − 6). Writing (a, b) and carrying both letters makes an easy question hard.
Distrusting a fractional centre.
Try 3 has its centre at (−5/2, 5/2). Nothing has gone wrong — centres do not have to be whole numbers.
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