Circles JEE Main 2026: Complete Guide & Strategy

Circles is a high-weightage chapter in JEE Main Mathematics, consistently contributing two to three questions per session. The chapter tests a well-defined set of concepts — the equation of a circle, tangent and normal conditions, chord-related results, the radical axis, and the family of circles — and rewards students who have built these concepts to an automaticity where they can apply them quickly. This guide covers every JEE-relevant concept with the exact problem-solving approaches that earn marks efficiently.

Equation of a Circle

The standard equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r². The general second-degree equation x² + y² + 2gx + 2fy + c = 0 represents a circle with center (-g, -f) and radius √(g² + f² - c), provided g² + f² - c > 0. Converting between standard and general forms, and extracting center and radius from the general form, are foundational skills tested in every examination.

JEE Main tests the conditions for a circle to pass through given points, to be tangent to a given line, or to have its center on a given line or curve. Problems often combine two or three such conditions simultaneously, requiring you to set up a system of equations. The parametric form (x = h + r cos θ, y = k + r sin θ) is useful for problems about points on a circle and is tested in problems about chords and tangents from a point. Connect with our straight lines guide for the line geometry that underlies many circle problems.

Tangents and Normals to Circles

The tangent to a circle at a point on the circle is perpendicular to the radius at that point. The equation of the tangent to x² + y² + 2gx + 2fy + c = 0 at the point (x₁, y₁) on the circle is obtained by replacing x² with xx₁, y² with yy₁, x with (x+x₁)/2, and y with (y+y₁)/2 in the circle equation — this T = 0 substitution rule is a key JEE technique. The condition for a line y = mx + c to be tangent to a circle is that the distance from the center to the line equals the radius.

The length of the tangent from an external point (x₁, y₁) to the circle is √(x₁² + y₁² + 2gx₁ + 2fy₁ + c). This formula is used in problems about tangent lengths and chord of contact. From an external point, two tangents can be drawn to a circle; they are equal in length. Take a free mock test on circles to practice tangent calculations quickly.

Chord of Contact and Power of a Point

If two tangents are drawn from an external point (x₁, y₁) to a circle, the chord joining the two points of tangency is called the chord of contact. Its equation is found by the same T = 0 substitution. The power of a point with respect to a circle is the square of the tangent length from that point, and it equals the product of the signed lengths in which any chord through the point divides. If the power is negative, the point is inside the circle.

The chord joining two points with the same parameter in the parametric form, the common chord of two circles, and the locus of points from which a given chord subtends a right angle at the center are all standard JEE Main problem types. The common chord of two circles is found by subtracting their equations.

Radical Axis and Family of Circles

The radical axis of two circles is the locus of points with equal power with respect to both circles. It is always a straight line, perpendicular to the line joining the centers. The radical axis of two intersecting circles is their common chord. The radical center of three circles is the point where all three pairwise radical axes meet. JEE Main tests the equations of radical axes and the determination of the radical center.

The family of circles passing through the intersection of two given circles S₁ = 0 and S₂ = 0 is given by S₁ + λS₂ = 0 for arbitrary λ. This family includes the radical axis as a degenerate case (λ = -1). Problems using the family of circles to find the specific circle satisfying an additional condition are a reliable JEE Main question type. For the conic sections that build on circle geometry, see our conic sections guide. For a complete coordinate geometry strategy, follow our coordinate geometry guide and upgrade for ₹149/month for our circles problem bank.