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Circles revision notes

Every idea in the Circles chapter for JEE Main, with the usual mistakes and the fast methods. 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 concept, and "last" is the latest year one appeared. ★ marks ideas 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 at origin)(x-h)^2+(y-k)^2=r^2,\qquad x^2+y^2=r^2\ \text{(centre at origin)}
Centre (h,k)(h,k), radius r>0r>0. A point is on the circle iff its distance from the centre is exactly rr.
General form
10× asked, last 2026
x2+y2+2gx+2fy+c=0,centre (−g,−f),r=g2+f2−cx^2+y^2+2gx+2fy+c=0,\qquad \text{centre }(-g,-f),\qquad r=\sqrt{g^2+f^2-c}
Needs equal x2x^2, y2y^2 coefficients and no xyxy term. Real iff g2+f2−c≥0g^2+f^2-c\ge0; equality gives a point circle.
xxyy(−g,−f)=(2,−1)(-g,-f)=(2,-1)r=g2+f2−c=3r=\sqrt{g^2+f^2-c}=3
x2+y2−4x+2y−4=0x^2+y^2-4x+2y-4=0: g=−2g=-2, f=1f=1, c=−4c=-4, so centre (2,−1)(2,-1), radius 3
  • Not g,f,cg,f,c read straight off ax2+ay2+⋯=0ax^2+ay^2+\dots=0. Divide by aa first. A condition found on the reduced constant c/ac/a must be multiplied back by aa before answering.
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
Circle with the segment from (x1,y1)(x_1,y_1) to (x2,y2)(x_2,y_2) as diameter: every point on it sees the diameter at 90∘90^\circ.
xxyyA(4,0)A(4,0)B(0,3)B(0,3)PP
∠AOB=90∘\angle AOB=90^\circ, so ABAB is a diameter: x2+y2−4x−3y=0x^2+y^2-4x-3y=0, AB=2r=5AB=2r=5; every PP on it sees ABAB at 90∘90^\circ
  • A circle through the origin that cuts the axes again at A(a,0)A(a,0) and B(0,b)B(0,b) has ABAB as a diameter (∠AOB=90∘\angle AOB=90^\circ): x2+y2−ax−by=0x^2+y^2-ax-by=0 and AB=2rAB=2r.
Parametric form
5× asked, last 2026
x=h+rcos⁡θ,y=k+rsin⁡θ,0≤θ<2πx=h+r\cos\theta,\qquad y=k+r\sin\theta,\qquad 0\le\theta<2\pi
θ\theta: angle of the radius from the +x+x direction. An arc 1/n1/n of the perimeter subtends 2π/n2\pi/n at the centre.
xxyyrrθ\theta(h,k)(h,k)(h+rcos⁡θ, k+rsin⁡θ)(h+r\cos\theta,\,k+r\sin\theta)
θ\theta is the angle of the radius from the +x+x direction
  • Quick method. Locus of (p+acos⁡t+bsin⁡t, q+asin⁡t−bcos⁡t)(p+a\cos t+b\sin t,\ q+a\sin t-b\cos t): subtract pp and qq, square and add; the cross terms cancel, giving a circle of radius a2+b2\sqrt{a^2+b^2} about (p,q)(p,q).
Circle through three points
3× asked, last 2025
put each point in x2+y2+2gx+2fy+c=0, solve for g,f,c\text{put each point in } x^2+y^2+2gx+2fy+c=0,\ \text{solve for } g,f,c
Points must not be collinear. Through the origin: c=0c=0 at once. The centre lies on the perpendicular bisector of every chord.
(0,0)(0,0)(4,0)(4,0)(1,3)(1,3)centre
The centre is on the perpendicular bisector of every chord: two of them fix it
  • Not every root of the fourth-point equation. The points are distinct. (2k,3k)(2k,3k) on x2+y2−x−y=0x^2+y^2-x-y=0 (which passes through the given origin) gives 13k2=5k13k^2=5k; k=0k=0 is the origin again, so k=5/13k=5/13.
Mirror image of a circle
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 (x1,y1)(x_1,y_1) in ax+by+c=0ax+by+c=0. The radius is unchanged, so equate the two radii.
xxyyy=x+1y=x+1(3,12)(3,\frac12)(−12,4)(-\frac12,4)
Reflect only the centre (here (x,y)→(y−1, x+1)(x,y)\to(y-1,\,x+1)); the radius stays
  • Quick method. Mirror in y=x+cy=x+c: (x,y)→(y−c, x+c)(x,y)\to(y-c,\ x+c). In x+y=cx+y=c: (x,y)→(c−y, c−x)(x,y)\to(c-y,\ c-x). If a line is the perpendicular bisector of PQPQ, then QQ is the image of PP in it.

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14 more concepts 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

  • Intercepts on the axes5×
  • Circle touching the axes12×
  • Position of a point (power)4×
  • Nearest and farthest points7×

Line and circle; chords

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

Two circles

  • Position of two circles11×
  • Point of contact of touching circles3×
  • Common chord and radical axis5×
  • Family of circles1×

Loci, figures and areas

  • Loci linked to a circle4×
  • Inscribed figures and areas7×

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