Circles
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Direct answer
The circle x^2 + y^2 + 2gx + 2fy + c = 0 has centre (-g, -f) and radius sqrt(g^2 + f^2 - c), real only when g^2 + f^2 - c > 0. The tangent at a point (x1, y1) on the circle follows the T = 0 rule — xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 — and the tangent length from an external point is the square root of the circle's expression evaluated there.
What you must remember
- Centre and radius: from x^2 + y^2 + 2gx + 2fy + c = 0, centre (-g, -f) and radius sqrt(g^2 + f^2 - c); x^2 + y^2 = r^2 is the centre-origin special case.
- Parametric point: (h + r cos theta, k + r sin theta) on a circle with centre (h, k), radius r.
- Tangents: at (x1, y1) on x^2 + y^2 = r^2 the tangent is xx1 + yy1 = r^2; a line y = mx + c touches this circle when c = ± r sqrt(1 + m^2).
- Tangent length: from an external point (x1, y1) it is sqrt(S1), where S1 is the circle's expression evaluated at the point; the two tangents from a point have equal lengths.
- Position tests: a point lies outside, on or inside as S1 is positive, zero or negative; a line cuts, touches or misses by comparing its distance from the centre with r; two circles by the distance between centres versus the sum and difference of radii.
- Family of circles: S + lambda L = 0 runs through the intersection of S = 0 with L = 0; S1 + lambda S2 = 0 through the common points of two circles.
- Orthogonal circles: they cut at right angles when 2g1g2 + 2f1f2 = c1 + c2.
Common confusion
Tangent at a point versus tangent from a point. The T = 0 formula is valid only when (x1, y1) lies on the circle; applied to an external point it gives the chord of contact — the line joining the two touch points — not a tangent. Also remember the count: two tangents from outside, one from the circle itself, none from inside.
Exam-focused takeaway
JEE Main asks centre-radius conversions, tangent equations, tangent lengths and position tests — quick, formula-direct marks. JEE Advanced adds the common-chord and radical-axis ideas, the family trick S + lambda L chosen from an extra condition, and circle loci from a moving point. The reliable routine: write both circles in general form, compute centres and radii, then interpret — most errors are sign slips in reading off g and f.
Frequently asked questions
What are the centre and radius of x^2 + y^2 + 2gx + 2fy + c = 0?
Centre (-g, -f) and radius sqrt(g^2 + f^2 - c), real only when g^2 + f^2 - c is positive.
How many tangents can be drawn from a point?
Two from an external point, one from the circle, none from inside; the two external tangents have equal lengths sqrt(S1).
What is the chord of contact?
For an external point, the line joining the two points of tangency, given by the same T = 0 expression.
When are two circles orthogonal?
When 2g1g2 + 2f1f2 = c1 + c2; the tangents at each intersection then meet at right angles.
What is the family S + lambda L = 0?
The set of all circles through the intersection of circle S = 0 and line L = 0; fix lambda from an extra condition.