Circles centred at the origin

OCR J560 Higher (J560/04–06) tests the equation of a circle centred at the origin and the equation of a tangent at a given point. This is a major problem-solving topic.

Equation of a circle centred at origin

A circle centred at (0, 0) with radius r has equation:

x² + y² = r²

Examples:

• x² + y² = 25 has centre (0, 0) and radius 5.
• x² + y² = 7 has centre (0, 0) and radius √7.

The radius is always the square root of the right-hand side. If the equation reads x² + y² = 49, then r = 7, NOT 49.

Determining whether a point lies on the circle

Substitute (a, b) into x² + y². If x² + y² = r² with the equation, the point is on the circle. If less than r², the point is inside; if greater, outside.

Tangent at a point on the circle

Key fact: a tangent to a circle at point P is perpendicular to the radius from the centre to P.

Method to find the tangent equation at P = (a, b) on x² + y² = r²:

1. Gradient of radius OP = b/a.
2. Gradient of tangent = perpendicular = −a/b (negative reciprocal).
3. Use point-slope form: y − b = (−a/b)(x − a).
4. Rearrange to a clean form, e.g. ax + by = r².

So the standard tangent equation at (a, b) is:

ax + by = r²

(where the circle is x² + y² = r²)

OCR mark scheme conventions

• M1 for stating the gradient of the radius.
• M1 for taking the negative reciprocal.
• M1 for using point-slope form with the given point.
• A1 for the tangent equation in a clean form.
• "Find the equation" — the answer must be an equation, not a gradient.