Circles & Systems of Circles
SRMJEEE – Mathematics
(Self‑Review & Mastery – JEE Advanced Equivalent)
1. Quick Overview
| Item | What it is | Key Formula |
|---|---|---|
| Circle | All points at a fixed distance (r) from a centre ((h,k)). | ((x-h)^2+(y-k)^2=r^2) or (x^2+y^2+2gx+2fy+c=0) where (g=-h,;f=-k,;c=h^2+k^2-r^2). |
| General Circle | Standard form with no restrictions on (g,f,c). | (x^2+y^2+2gx+2fy+c=0). |
| Radius | Distance from centre to any point on the circle. | (r=\sqrt{h^2+k^2-c}). |
| Tangency | Two circles touch at exactly one point. | ( |
| Intersection | Two circles cut at two points. | (d < r_1 + r_2) and (d > |
| Radical Axis | Locus of points with equal power w.r.t two circles. | Line: ((x^2+y^2+2g_1x+2f_1y+c_1) - (x^2+y^2+2g_2x+2f_2y+c_2)=0). |
| Coaxial Circles | Family of circles sharing the same radical axis. | (S_1 + \lambda S_2 = 0) (linear combination of two circle equations). |
Tip: Convert everything to the general form before manipulating; it keeps arithmetic uniform.
2. Key Concepts & Properties
2.1 Circle Equation Forms
| Form | When to use | Remarks |
|---|---|---|
| ((x-h)^2+(y-k)^2=r^2) | Given centre or radius | Easy to read, intuitive |
| (x^2+y^2+2gx+2fy+c=0) | Algebraic manipulation, intersection, radical axis | (g=-h,;f=-k,;c=h^2+k^2-r^2) |
2.2 Distance Between Two Circle Centres
For circles (S_1: (x-h_1)^2+(y-k_1)^2=r_1^2) and (S_2: (x-h_2)^2+(y-k_2)^2=r_2^2):
[ d = \sqrt{(h_1-h_2)^2+(k_1-k_2)^2} ]
2.3 Conditions for Tangency / Intersection
| Relationship | Condition |
|---|---|
| Externally tangent | (d = r_1 + r_2) |
| Internally tangent | (d = |
| Intersecting (2 pts) | ( |
| Separate (no intersection) | (d > r_1+r_2) |
| Coincident | (h_1=h_2,; k_1=k_2,; r_1=r_2) |
2.4 Radical Axis
- Locus of points having equal power w.r.t two circles.
- Equation: (S_1 - S_2 = 0) where (S_i) are general forms of circles.
- It is a straight line unless the circles are concentric, in which case it is undefined.
2.5 Coaxial System
- A set of circles sharing the same radical axis.
- General member: (S_1 + \lambda S_2 = 0), (\lambda\in\mathbb{R}).
3. Working with Systems of Circles
3.1 Finding a Common Tangent
Given two circles, the common tangents can be:
- External: Do not intersect the segment joining centres.
- Internal: Intersect that segment.
Method
- Compute (d), (r_1), (r_2).
- Use formula for length of tangent from an external point or use power of a point.
- For perpendicular tangents, set up equations of lines (y = mx + c) and impose conditions that the distance from each centre to the line equals the radius.
3.2 Intersection Points
Solve the system: [ \begin{cases} (x-h_1)^2+(y-k_1)^2=r_1^2\ (x-h_2)^2+(y-k_2)^2=r_2^2 \end{cases} ] Subtract one from the other to get a linear equation (the radical axis), then substitute back.
3.3 Constructing a Circle with Given Conditions
Common situations:
- Through three points: Solve for ((h,k,r)) using the three point equations.
- Tangent to a line: Use distance from centre to line = radius.
- Passing through a point and tangent to another circle: Combine power-of-point condition with tangency condition.