Circles: Tangents, Chords & Radical Axis JEE Main Guide

Circles is one of the most rewarding chapters in JEE Main Coordinate Geometry — the number of JEE-targetted results per page of study is high, and most questions are direct applications of five to eight standard results rather than novel derivations. Students who internalise the tangent, chord, and radical axis formulae in their general forms can answer circle questions in 90 seconds. This guide gives you that standard form mastery.

Circle Equations and the General Form

Standard form: (x−h)² + (y−k)² = r² (centre (h,k), radius r). General form: x² + y² + 2gx + 2fy + c = 0 (centre (−g,−f), radius √(g²+f²−c)). The intercept forms: x-intercept = 2√(g²−c) (set y=0), y-intercept = 2√(f²−c). For two circles to intersect: |r₁−r₂| < d < r₁+r₂ where d is the distance between centres. Touching externally: d = r₁+r₂. Touching internally: d = |r₁−r₂|. These intersection conditions are tested once per three sessions as a True/False or matching option.

The condition for a point (x₁,y₁) to lie on, inside, or outside a circle: S₁ = x₁²+y₁²+2gx₁+2fy₁+c. S₁ < 0: inside; S₁ = 0: on; S₁ > 0: outside. This notation S₁ is standard in all subsequent tangent and chord formulae — master it first. Test your Circles speed with a free mock. For the complete Coordinate Geometry framework, see our coordinate geometry guide.

Tangent Equations: The T = 0 Rule

Equation of tangent at a point (x₁,y₁) on the circle x²+y²+2gx+2fy+c=0: T = 0 where T ≡ xx₁+yy₁+g(x+x₁)+f(y+y₁)+c. The "T" notation is the most efficient tool in circle geometry: replace x² by xx₁, y² by yy₁, x by (x+x₁)/2, and y by (y+y₁)/2. For the standard circle x²+y²=r²: tangent at (x₁,y₁) is xx₁+yy₁=r². Condition for a line y=mx+c to be tangent to x²+y²=r²: c² = r²(1+m²), giving c = ±r√(1+m²). Length of tangent from external point (x₁,y₁) to circle: √S₁ = √(x₁²+y₁²+2gx₁+2fy₁+c).

Common tangents to two circles: external common tangents (do not pass between circles) — divide the line joining centres externally in ratio r₁:r₂; internal common tangents (pass between, circles external to each other) — divide the line joining centres internally in ratio r₁:r₂. Number of common tangents: 4 (circles external), 3 (one touches other externally), 2 (one inside other with common external), 1 (internally tangent), 0 (one inside other non-touching). For circles within Coordinate Geometry broader context see our circles complete guide.

Chord of Contact and Chord with Given Midpoint

Chord of contact: If tangents are drawn from external point (x₁,y₁) to circle x²+y²+2gx+2fy+c=0, the chord of contact (the line joining the two tangent points) has equation T=0 (the same T expression as the tangent formula!). This remarkable result means: the equation of the chord of contact from (x₁,y₁) is T=xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0. JEE uses this directly in "find the chord of contact from the point (2,3) to the circle..."

Equation of a chord with given midpoint (h,k): T = S₁ where T is the chord formula evaluated at general (x,y) with midpoint (h,k), and S₁ = h²+k²+2gh+2fk+c. Explicitly: x·h+y·k+g(x+h)+f(y+k)+c = h²+k²+2gh+2fk+c, which simplifies. The slope of this chord: m = −(h+g)/(k+f). This result is used in the "chord bisected at a given point" type JEE question, which appears about once per year. For the chord-pole-polar family of results, see our circle family and radical axis guide.

Power of a Point and Radical Axis

Power of a point P with respect to a circle: π = d² − r² where d is the distance from P to the centre and r is the radius. If P is inside: π < 0; on circle: π = 0; outside: π = S₁ (the same as squared tangent length). Radical axis of two circles (locus of points with equal power with respect to both circles): S₁ = S₂, which simplifies to a straight line (the equation of the radical axis is S₁ − S₂ = 0). Three circles have a common radical axis meeting point called the radical centre. These concepts appear in JEE as direct questions and as options in multiple correct format. For related locus techniques, see our locus problems guide.