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Circles formula sheet

Every formula in the Circles chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 20 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Equation of a circle

Centre–radius form
2× asked, last 2025
(x−h)2+(y−k)2=r2,x2+y2=r2 (centre O)(x-h)^2+(y-k)^2=r^2,\qquad x^2+y^2=r^2\ (\text{centre O})
Centre (h,k)(h,k), radius r>0r>0.
General form
10× asked, last 2026
x2+y2+2gx+2fy+c=0centre (−g,−f),r=g2+f2−c\begin{aligned}&x^2+y^2+2gx+2fy+c=0\\&\text{centre }(-g,-f),\qquad r=\sqrt{g^2+f^2-c}\end{aligned}
Equal x2x^2, y2y^2 coefficients, no xyxy; divide by aa first. g2+f2−c=0g^2+f^2-c=0: point circle.
Diametric form
5× asked, last 2026
(x−x1)(x−x2)+(y−y1)(y−y2)=0(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0
Through O cutting axes at (a,0)(a,0), (0,b)(0,b): x2+y2−ax−by=0x^2+y^2-ax-by=0, AB=2rAB=2r.
Parametric form
5× asked, last 2026
x=h+rcos⁡θ,y=k+rsin⁡θx=h+r\cos\theta,\qquad y=k+r\sin\theta
(p+acos⁡t+bsin⁡t, q+asin⁡t−bcos⁡t)(p+a\cos t+b\sin t,\ q+a\sin t-b\cos t): circle of radius a2+b2\sqrt{a^2+b^2} about (p,q)(p,q).
Through three points
3× asked, last 2025
substitute each point in x2+y2+2gx+2fy+c=0\text{substitute each point in }x^2+y^2+2gx+2fy+c=0
Non-collinear. Through O: c=0c=0. Centre on every chord's perpendicular bisector.
Mirror image
5× asked, last 2026
x−x1a=y−y1b=−2(ax1+by1+c)a2+b2\frac{x-x_1}a=\frac{y-y_1}b=\frac{-2(ax_1+by_1+c)}{a^2+b^2}
Image of the centre in ax+by+c=0ax+by+c=0; radius unchanged. In y=x+cy=x+c: (x,y)→(y−c, x+c)(x,y)\to(y-c,\ x+c).

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14 more formulas in 4 topics

Axes, points and distances; Line and circle; chords; Two circles; and Loci, figures and areas. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Axes, points and distances

  • Intercepts5×
  • Touching the axes12×
  • Power of a point4×
  • Nearest and farthest7×

Line and circle; chords

  • Line and circle10×
  • Circle touching lines6×
  • Length of a chord13×
  • Chord with given midpoint8×

Two circles

  • Position of two circles11×
  • Point of contact3×
  • Common chord, radical axis5×
  • Family of circles1×

Loci, figures and areas

  • Locus of a section point4×
  • Inscribed figures7×

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About 120 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

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Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

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