Circles - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Find the Equation of a Circle

Pattern

Find the equation of a circle given center and radius, or given three points, or given end-points of a diameter

How to recognize

Asks for the equation of a circle passing through given points, or with a given center and radius, or touching a given line.

→ Standard form: (x-h)^2 + (y-k)^2 = r^2. For three points, substitute into the general equation and solve for g, f, c. For diameter end-points, use (x-x1)(x-x2) + (y-y1)(y-y2) = 0.

Tangent and Normal to a Circle

Pattern

Find the equation of tangent or normal to a circle at a given point or with a given slope

How to recognize

Asks for the tangent or normal at a specific point on the circle, or tangent with a given slope, or tangent from an external point.

→ At point (x1, y1) on the circle: use T = 0. For tangent with slope m to x^2 + y^2 = a^2: y = mx +/- a*sqrt(1+m^2). From external point: use the condition for tangency and solve for m.

Chord Properties

Pattern

Find chord length, chord of contact, midpoint of chord, or equation of chord with given midpoint

How to recognize

Mentions chord, chord of contact, midpoint of chord, or chord bisected at a point.

→ Chord of contact from (x1,y1): T = 0. Chord with midpoint (x1,y1): T = S1. Chord length: 2*sqrt(r^2 - d^2) where d is perpendicular distance from center to chord.

Family of Circles

Pattern

Find a circle through intersection of two circles or through intersection of a circle and a line

How to recognize

Asks for a circle passing through the intersection points of two given circles or a circle and a line, with an additional condition.

→ Family through intersection of S1 and S2: S1 + lambda*S2 = 0. Family through intersection of circle S and line L: S + lambda*L = 0. Use the additional condition to find lambda.

Radical Axis Problems

Pattern

Find the radical axis, radical center, or use power of a point

How to recognize

Mentions radical axis, radical center, or equal tangent lengths from a point to two circles.

→ Radical axis: S1 - S2 = 0. Radical center: intersection of pairwise radical axes. The radical axis is perpendicular to the line of centers.

Power of a Point

Pattern

Find the power of a point with respect to a circle or use it to determine position

How to recognize

Asks whether a point is inside/outside/on a circle, or asks for the power or length of tangent.

→ Power = S1 (substitute the point into the circle equation). S1 > 0: outside, S1 = 0: on circle, S1 < 0: inside. Length of tangent = sqrt(S1).

Orthogonal Circles

Pattern

Check orthogonality or find a circle orthogonal to given circles

How to recognize

Asks whether two circles cut orthogonally, or to find a circle orthogonal to given circles.

→ Use the condition 2g1*g2 + 2f1*f2 = c1 + c2. Alternatively, at the intersection point, the tangent to one circle passes through the center of the other.

Parametric Form Problems

Pattern

Use parametric representation to find points on a circle satisfying given conditions

How to recognize

Involves parametric coordinates (a*cos(theta), a*sin(theta)) or asks to parameterize a point on the circle.

→ Write points as (h + r*cos(theta), k + r*sin(theta)). Apply the given condition to find theta. Common in tangent-normal problems and locus problems.

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3D Geometry

Inverse Trigonometric Functions

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