8. Coordinate geometry of the circle
- Syllabus
- 0606–2028–2029
- Topic
- 8
- Level
- —
Learning objectives
8.1Use circle equations• Know and use the equation of a circle with radius r and centre (a, b).• Identify the centre and radius from circle equations in any form, including (x - a)^2 + (y - b)^2 = r^2 and x^2 + y^2 + 2gx + 2fy + c = 0.• The formula is given in the List of formulas.8.2Solve line-circle intersection problems• Solve problems involving the intersection of a circle and a straight line.• Find points of intersection and determine whether a line is a tangent, a chord or does not intersect the circle.8.3Solve tangent problems• Solve problems involving tangents to a circle, including finding equations of tangents.• No use of calculus is expected.8.4Solve circle-circle intersection problems• Solve problems involving the intersection of two circles.• Find points of intersection, find the equation of a common chord, and determine whether two circles intersect, touch or do not intersect.
Read and construct circle equations
| Equation | Centre | Radius |
|---|---|---|
| (x−a)2+(y−b)2=r2 | (a,b) | r |
| x2+y2+2gx+2fy+c=0 | (−g,−f) | g2+f2−c |
The standard form is the distance formula from a general point (x,y) to the fixed centre (a,b). To convert a general equation, complete each square using x2+2gx=(x+g)2−g2 and the matching identity for y.
If A(−3,6) and B(7,−8) are endpoints of a diameter, the centre is their midpoint (2,−1). The radius squared is (−3−2)2+(6+1)2=74, so the circle is (x−2)2+(y+1)2=74.
To classify a point, compare its squared distance from the centre with r2: smaller means inside, equal means on the circumference, larger means outside. In standard form, the centre signs are opposite to those visible inside the brackets.
Classify a line's intersection with a circle
Substitute the line equation into the circle equation. The resulting quadratic describes the possible intersection coordinates; back-substitute every real root to obtain the full points.
| Discriminant Δ=b2−4ac | Real roots | Geometry |
|---|---|---|
| Δ>0 | two | the line cuts a chord |
| Δ=0 | one repeated root | the line is tangent |
| Δ<0 | none | the line does not meet the circle |
For (x−5)2+(y−2)2=5 and y=2x−3, substitution simplifies to 5x2−30x+45=5(x−3)2=0. There is one repeated root, so the line is tangent; x=3 gives the contact point (3,3).
An axis is just a special line: use x=0 for the y-axis and y=0 for the x-axis. For (x−4)2+(y−2)2=40 on the y-axis, (y−2)2=24, giving (0,2±26).
Do not classify from a sketch alone. A repeated quadratic root is one coordinate value, but the point of contact still needs the other coordinate from the line.
Find a circle tangent without calculus
The tangent at a point on a circle is perpendicular to the radius through that point. This right-angle fact supplies the tangent gradient; calculus is not required.
Identify the centre, verify or use the contact point, calculate the radius gradient, take its negative reciprocal for the tangent gradient, then use point-gradient form through the contact point.
A circle has centre (4,−3) and contains A(3,−1). The radius CA has gradient (−1+3)/(3−4)=−2, so the tangent gradient is 1/2. Hence the tangent is y+1=21(x−3).
The tangent passes through the point on the circumference, not through the centre. A horizontal radius gives a vertical tangent; a vertical radius gives a horizontal tangent, so the negative-reciprocal formula is not used with an undefined gradient.
Compare and intersect two circles
For circles with centre distance d and radii r1,r2, compare d with r1+r2 and ∣r1−r2∣. These distances describe when the circles can reach each other externally or internally.
| Condition | Relationship |
|---|---|
| d>r1+r2 | separate outside |
| d=r1+r2 | touch externally |
| ∣r1−r2∣<d<r1+r2 | intersect at two points |
| d=∣r1−r2∣ | touch internally |
| d<∣r1−r2∣ | one lies inside the other without touching |
To find common intersection points, subtract the two expanded circle equations. The x2 and y2 terms cancel, leaving the straight-line equation of the common chord. Solve that line simultaneously with either original circle.
Centres (8,5) and (10,6.5) are 22+1.52=2.5 apart. With radii 4 and 1.5, ∣4−1.5∣=2.5, so the circles touch internally; no contact coordinate is needed for this conclusion.
Subtracting circle equations gives the common-chord line, not the intersection points by itself. If d=0 and r1=r2, the equations describe the same circle rather than one of the five distinct-circle cases above.