Shifted circles, chords and tangents
| English | 中文 | Pinyin |
|---|---|---|
| centre/ˈsentə/ | 圆心 | yuán xīn |
Where is the centre hidden in an expanded circle equation?
- An expanded circle equation can hide its centre and radius. Completing the square exposes the geometry needed for a chord or tangent.
- This lesson studies centre 圆心: The point at equal distance from every point on a circle.
Choose the mathematical structure
- Rewrite a circle as (x−a)²+(y−b)²=r² by completing both squares. A perpendicular from the centre bisects a chord, so half-chord length follows from a right triangle. A tangent is perpendicular to the radius at the contact point. An angle subtended by a diameter at the circumference is 90°.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines centre?
The point at equal distance from every point on a circle.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x²+y²−4x+2y−20=0, completing squares gives (x−2)²+(y+1)²=25: centre (2,−1), radius 5. The chord y=2 is 3 units above the centre; its half-length is √(25−9)=4, so endpoints are (−2,2),(6,2) and length 8. At P(5,3), the radius gradient is 4/3, giving tangent y−3=−3(x−5)/4 or 3x+4y=27. For diameter endpoints A(−3,−1), B(7,−1) and circumference point T(2,4), gradients AT=1 and BT=−1 verify angle ATB=90°.
Shifted circles, chords and tangents
Rewrite a circle as (x−a)²+(y−b)²=r² by completing both squares
Match each coordinate calculation to its geometric or contextual condition.
Find the radius of x²+y²−4x+2y−20=0.
The completed equation has r²=4+1+20=25, hence r=5.
Test a tempting shortcut
- Completing a square adds a constant: adjust the other side too. The circle’s centre has the opposite signs to the bracket constants. A proposed tangent point must lie on the circle; perpendicularity alone at an unrelated point is insufficient.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A line perpendicular to any radius is automatically a tangent, wherever it crosses the plane. This claim is false. Explain which definition or assumption it violates.
Find the length of its chord y=2.
Half-length=√(25−3²)=4, so full chord length is 8.
A line perpendicular to any radius is automatically a tangent, wherever it crosses the plane.
Completing a square adds a constant: adjust the other side too. The circle’s centre has the opposite signs to the bracket constants. A proposed tangent point must lie on the circle; perpendicularity alone at an unrelated point is insufficient.
Interpret a new situation
- Check radius distance for each contact point and both chord endpoints. A negative completed r² gives no real circle, while r²=0 gives a single point rather than a nondegenerate circle. Handle a horizontal or vertical radius without dividing by zero.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
At P(5,3), find c in the tangent equation 3x+4y=c.
At P, c=3×5+4×3=27.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · C. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The point at equal distance from every point on a circle. Choose the relationship, show the method, check its assumptions and interpret the result.