Circles centred at the origin
This is exclusively a Higher topic on Edexcel 1MA1, examined on Paper 1H. The skill: recognise the equation x² + y² = r², find the radius, and find the equation of the tangent at a given point.
Standard equation
A circle centred at the origin (0, 0) with radius r has equation:
x² + y² = r²
So x² + y² = 25 is a circle centred at origin with radius 5.
Recognising radius
Take the square root of the constant:
- x² + y² = 9 → r = 3.
- x² + y² = 50 → r = √50 = 5√2.
Tangent at a point on the circle
At a point P(a, b) on the circle x² + y² = r², the tangent line is perpendicular to the radius OP.
Steps:
- Compute the gradient of OP: m_OP = b / a.
- The gradient of the tangent is m_T = −a / b (negative reciprocal).
- Use y − b = m_T (x − a) to write the tangent equation.
Equivalent compact form: ax + by = r² (the point-tangent form for circles centred at origin).
✦Worked example— Example
Circle: x² + y² = 25. Point P(3, 4).
- Check P is on the circle: 9 + 16 = 25 ✓.
- Tangent: 3x + 4y = 25.
Common Edexcel mark-scheme phrasing
- B1 for the radius.
- M1 for the gradient of the radius.
- M1 for the perpendicular gradient.
- M1 for the substitution into y − b = m(x − a).
- A1 for the tangent equation in the required form.
⚠Common mistakes— Common errors
- Stating r² instead of r.
- Using the gradient of the radius for the tangent (forgetting to take the negative reciprocal).
- Sign errors when using y − b = m(x − a).
- Not checking the given point lies on the circle.
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