Shifted circles, chords and tangents
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| centre/ˈsentə/ | tâm đường tròn |
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.
The method
- 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°.
For x²+y²-4x+2y-20=0, why is the centre (2,-1) rather than (-4,2)?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Halve the linear coefficients to complete each square: (x-2)² and (y+1)². The centre reads (2,-1) - the bracket constants carry reversed signs - and the radius² is 20+4+1=25.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The completed equation has r²=4+1+20=25, hence r=5.
Where it goes wrong
- 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.
Not true: A line perpendicular to any radius is a tangent, wherever it crosses the plane. A tangent must be perpendicular to the radius at the point where it touches the circle.
Find the length of its chord y=2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Half-length=√(25−3²)=4, so full chord length is 8.
A line perpendicular to any radius is a tangent, wherever it crosses the plane.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A tangent must be perpendicular to the radius at the point where it touches the circle.
Applying it
- 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.
At P(5,3), find c in the tangent equation 3x+4y=c.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
At P, c=3×5+4×3=27.
Match each circle property to its consequence.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Half-chord length comes from the right triangle with the radius as hypotenuse; a tangent must be checked to touch the circle AT its stated point.
In your course
- AQA A-level Mathematics (7357), specification section C.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Centre: The point at equal distance from every point on a circle.
- Rewrite a circle as (x−a)²+(y−b)²=r² by completing both squares.
- Watch for: completing a square adds a constant: adjust the other side too.