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Circle formulas

Master Circle through 34 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Circle, every formula

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Standard equation (centre, radius)
$(x - α)^{2} + (y - β)^{2} = r^{2}$
centre (α,β), radius r
Q1MCQStandard equation
The equation of the circle with centre $(2,-3)$ and radius $4$ is:
- A
  $(x-2)^{2}+(y+3)^{2}=16$
- B
  $(x+2)^{2}+(y-3)^{2}=16$
- C
  $(x-2)^{2}+(y-3)^{2}=4$
- D
  $(x-2)^{2}+(y+3)^{2}=4$
Circle at origin
$x^{2} + y^{2} = r^{2}$
centre (0,0)
General equation
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
centre (-g,-f), radius \sqrt{g^2+f^2-c}
Q1MCQCentre of general circle
The centre of $x^{2}+y^{2}-4x+6y-3=0$ is:
- A
  $(2,-3)$
- B
  $(-2,3)$
- C
  $(4,-6)$
- D
  $(-4,6)$
Centre from general form
$C = (- g, - f)$
Q1MCQCentre coordinates
For $x^{2}+y^{2}+2gx+2fy+c=0$, the centre is:
- A
  $(-g,-f)$
- B
  $(g,f)$
- C
  $(-2g,-2f)$
- D
  $(g/2,f/2)$
Radius from general form
$r = g^{2} + f^{2} - c$
real circle needs g^2+f^2-c>0
Diameter form
$(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$
(x1,y1),(x2,y2) ends of a diameter
Q1MCQDiameter form
The circle on the segment from $(1,2)$ to $(3,4)$ as diameter is:
- A
  $(x-1)(x-3)+(y-2)(y-4)=0$
- B
  $(x+1)(x+3)+(y+2)(y+4)=0$
- C
  $(x-1)(x-3)-(y-2)(y-4)=0$
- D
  $x^{2}+y^{2}=4$
General 2nd-degree is a circle if
$a = b \neq = 0, h = 0$
coeff x^2 = coeff y^2, no xy term
Q1MCQCondition for a circle
$ax^{2}+by^{2}+2hxy+\cdots=0$ represents a circle if:
- A
  $a=b\neq0$ and $h=0$
- B
  $a=-b$
- C
  $h\neq0$
- D
  $a=b=0$
x-axis intercept
$∣ A B ∣ = 2 g^{2} - c$
for x^2+y^2+2gx+2fy+c=0
y-axis intercept
$∣ C D ∣ = 2 f^{2} - c$
Touches x-axis
$g^{2} = c$
Q1MCQTouches x-axis
The circle $x^{2}+y^{2}+2gx+2fy+c=0$ touches the x-axis if:
- A
  $g^{2}=c$
- B
  $f^{2}=c$
- C
  $g=0$
- D
  $c=0$
Touches y-axis
$f^{2} = c$
Q1MCQTouches y-axis
The circle touches the y-axis if:
- A
  $f^{2}=c$
- B
  $g^{2}=c$
- C
  $f=0$
- D
  $g^2=f^2$
Touches both axes
$g^{2} = f^{2} = c$
centre (±a,±a), radius a
Passes through origin
$c = 0$
Q1MCQThrough origin
$x^{2}+y^{2}+2gx+2fy+c=0$ passes through the origin iff:
- A
  $c=0$
- B
  $g=0$
- C
  $f=0$
- D
  $g=f$
Parametric point
$(α + r cos θ, β + r sin θ)$
0<=θ<2π
Q1MCQParametric point
A point on $x^{2}+y^{2}=9$ at parameter $\theta$ is:
- A
  $(3\cos\theta,3\sin\theta)$
- B
  $(9\cos\theta,9\sin\theta)$
- C
  $(\cos\theta,\sin\theta)$
- D
  $(3\sin\theta,3\cos\theta)$
Notation S11 (power of point)
$S_{11} = x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c$
Position of a point
$S_{11} > 0 outside, = 0 on, < 0 inside$
Q1MCQPosition of point
If $S_{11}<0$ for a point $P$, then $P$ lies:
- A
  inside the circle
- B
  on the circle
- C
  outside the circle
- D
  at the centre
Length of tangent from a point
$L = S_{11}$
P outside the circle
Length of chord (distance d from centre)
$2 r^{2} - d^{2}$
Tangent at (x1,y1) on S=0
$S_{1} \equiv x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$
Q1MCQTangent at a point
The tangent to $x^{2}+y^{2}=25$ at $(3,4)$ is:
- A
  $3x+4y=25$
- B
  $3x-4y=25$
- C
  $4x+3y=25$
- D
  $3x+4y=5$
Tangent at (x1,y1) on x^2+y^2=a^2
$x x_{1} + y y_{1} = a^{2}$
Q1MCQTangent form on x^2+y^2=a^2
The tangent at $(x_1,y_1)$ to $x^{2}+y^{2}=a^{2}$ is:
- A
  $xx_1+yy_1=a^{2}$
- B
  $xx_1-yy_1=a^{2}$
- C
  $x_1x+y_1y=0$
- D
  $xx_1+yy_1=a$
Tangent at parameter θ
$x cos θ + y sin θ = r$
on x^2+y^2=r^2
Q1MCQTangent at angle θ
The tangent to $x^{2}+y^{2}=r^{2}$ at parameter $\theta$ is:
- A
  $x\cos\theta+y\sin\theta=r$
- B
  $x\sin\theta+y\cos\theta=r$
- C
  $x\cos\theta-y\sin\theta=r$
- D
  $x\cos\theta+y\sin\theta=r^{2}$
Condition y=mx+c tangent to x^2+y^2=r^2
$c^{2} = r^{2} (1 + m^{2})$
Tangent of slope m
$y = m x \pm r 1 + m^{2}$
to x^2+y^2=r^2
Q1MCQTangent of slope m
A tangent of slope $m$ to $x^{2}+y^{2}=r^{2}$ is:
- A
  $y=mx\pm r\sqrt{1+m^{2}}$
- B
  $y=mx\pm r$
- C
  $y=mx\pm rm$
- D
  $y=mx\pm\sqrt{r^{2}+m^{2}}$
Line lx+my+n=0 tangent to x^2+y^2=r^2
$n^{2} = r^{2} (l^{2} + m^{2})$
Q1MCQLine tangent condition
$lx+my+n=0$ touches $x^{2}+y^{2}=r^{2}$ iff:
- A
  $n^{2}=r^{2}(l^{2}+m^{2})$
- B
  $n=r$
- C
  $l^{2}+m^{2}=r^{2}$
- D
  $n^{2}=l^{2}+m^{2}$
Normal to x^2+y^2=r^2
$x y_{1} - y x_{1} = 0$
passes through centre
Q1MCQNormal to circle
The normal to $x^{2}+y^{2}=r^{2}$ at $(x_1,y_1)$:
- A
  passes through the centre
- B
  is parallel to the tangent
- C
  never meets the circle
- D
  is $xx_1+yy_1=r^2$
Position of a line (d vs r)
$d > r no cut, d = r tangent, d < r two points$
Q1MCQLine vs circle
A line at distance $d$ from the centre with $d=r$ is:
- A
  a tangent
- B
  a secant
- C
  outside
- D
  a diameter
Chord of contact from P
$S_{1} = 0$
P external
Q1MCQChord of contact
The chord of contact of $P(x_1,y_1)$ w.r.t. $S=0$ is:
- A
  $S_1=0$
- B
  $S_{11}=0$
- C
  $S=0$
- D
  $S_{12}=0$
Polar of P
$S_{1} = 0$
polar line of pole P
Q1MCQPolar
The polar of a point $P(x_1,y_1)$ w.r.t. $S=0$ is:
- A
  $S_1=0$
- B
  $S=0$
- C
  $S_{11}=0$
- D
  $S_{12}=0$
Conjugate points
$S_{12} = 0$
Q1MCQConjugate points
$P$ and $Q$ are conjugate w.r.t. $S=0$ iff:
- A
  $S_{12}=0$
- B
  $S_{11}=0$
- C
  $S_1=0$
- D
  $S=0$
Director circle
$x^{2} + y^{2} = 2 r^{2}$
locus of ⟂ tangents to x^2+y^2=r^2
Q1MCQDirector circle
The director circle of $x^{2}+y^{2}=r^{2}$ is:
- A
  $x^{2}+y^{2}=2r^{2}$
- B
  $x^{2}+y^{2}=r^{2}/2$
- C
  $x^{2}+y^{2}=4r^{2}$
- D
  $x^{2}+y^{2}=r^{2}$
Angle between tangents from P
$tan \frac{θ}{2} = \frac{r}{S _{11}}$
Q1MCQAngle between tangents
The angle $\theta$ between the tangents from $P$ satisfies:
- A
  $\tan\dfrac{\theta}{2}=\dfrac{r}{\sqrt{S_{11}}}$
- B
  $\tan\theta=\dfrac{r}{S_{11}}$
- C
  $\sin\theta=\dfrac{r}{\sqrt{S_{11}}}$
- D
  $\tan\dfrac{\theta}{2}=\dfrac{\sqrt{S_{11}}}{r}$
Radical axis
$S - S_{1} = 0$
locus of equal power w.r.t. two circles
Q1MCQRadical axis
The radical axis of $S=0$ and $S_1=0$ is:
- A
  $S-S_1=0$
- B
  $S+S_1=0$
- C
  $S\cdot S_1=0$
- D
  $S_1=0$
Orthogonality condition
$2 g g_{1} + 2 f f_{1} = c + c_{1}$
two circles cut orthogonally
Length of common tangents
$direct d^{2} - (r_{1} - r_{2})^{2}, transverse d^{2} - (r_{1} + r_{2})^{2}$
d = distance between centres

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Circle formulas

Circle, every formula

Standard equation (centre, radius)

Circle at origin

General equation

Centre from general form

Radius from general form

Diameter form

General 2nd-degree is a circle if

x-axis intercept

y-axis intercept

Touches x-axis

Touches y-axis

Touches both axes

Passes through origin

Parametric point

Notation S11 (power of point)

Position of a point

Length of tangent from a point

Length of chord (distance d from centre)

Tangent at (x1,y1) on S=0

Tangent at (x1,y1) on x^2+y^2=a^2

Tangent at parameter θ

Condition y=mx+c tangent to x^2+y^2=r^2

Tangent of slope m

Line lx+my+n=0 tangent to x^2+y^2=r^2

Normal to x^2+y^2=r^2

Position of a line (d vs r)

Chord of contact from P

Polar of P

Conjugate points

Director circle

Angle between tangents from P

Radical axis

Orthogonality condition

Length of common tangents

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