Circle equations centred at the origin and tangent equations [Higher tier]
A circle centred at the origin with radius r has equation x² + y² = r². This comes directly from Pythagoras' theorem applied to a point (x, y) on the circle: the distance from the origin is r, so √(x² + y²) = r, square both sides.
Reading the equation
x² + y² = 25 → centre (0, 0), radius 5 (√25).
x² + y² = 49 → centre (0, 0), radius 7.
If the equation is given as x² + y² = 18, then r = √18 = 3√2 (in surd form).
Is a point on the circle?
Substitute the point's coordinates and check if both sides match.
Example: is (3, 4) on x² + y² = 25? 3² + 4² = 9 + 16 = 25 ✓ Yes.
Is (1, 5) on the same circle? 1 + 25 = 26 ≠ 25 — outside the circle.
Tangent at a point on the circle
The tangent at a point P on a circle is perpendicular to the radius at P.
Strategy:
- Find the gradient of the radius from the origin to P:
m_radius = y_P / x_P. - Take the negative reciprocal:
m_tangent = −x_P / y_P. - Write the equation using point-gradient form:
y − y_P = m_tangent (x − x_P).
✦Worked example— Worked example — tangent at (3, 4)
Circle x² + y² = 25 and point P(3, 4).
- Confirm on circle: 9 + 16 = 25 ✓.
- Radius gradient: 4/3.
- Tangent gradient: -3/4.
- Tangent equation:
y - 4 = -¾(x - 3). Multiply through by 4:4y - 16 = -3(x - 3) = -3x + 9 ⇒ 3x + 4y = 25.
So the tangent is 3x + 4y = 25 (or equivalently y = -¾x + 25/4).
Useful identity
For circle x² + y² = r² and point (x_P, y_P) on the circle, the tangent at P has equation:
x·x_P + y·y_P = r².
(Quick proof: starts from the perpendicular-radius gradient calculation and simplifies.)
For the worked example: 3x + 4y = 25 — same answer.
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting to take square root for r.
x² + y² = 25has radius 5, not 25. - Negative reciprocal sign error. Reciprocal of 4/3 is 3/4; negative reciprocal is -3/4.
- Using wrong perpendicular point. The tangent meets the circle at P, not at the origin.
- Equation only valid for circles centred at origin in GCSE. Off-centre circles are A-level material.
- Mixing up "is on the circle" with "is inside/outside". Equality means on;
<means inside,>means outside.
➜Try this— Quick check
Find the tangent to x² + y² = 13 at the point (2, 3).
Verify: 4 + 9 = 13 ✓. Use the shortcut: 2x + 3y = 13.
AI-generated · claude-opus-4-7 · v3-deep-algebra