Circle formulas

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Formulas: 34
Subject: Mathematics
Exam: JEE Advanced
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Circle, every formula

34 formulas, typeset and free. Print it, or keep it open beside your practice.

Standard equation (centre, radius)
$(x - α)^{2} + (y - β)^{2} = r^{2}$
centre (α,β), radius r
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}
Centre from general form
$C = (- g, - f)$
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
General 2nd-degree is a circle if
$a = b \neq = 0, h = 0$
coeff x^2 = coeff y^2, no xy term
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
$g^{2} = c$
Touches y-axis
$f^{2} = c$
Touches both axes
$g^{2} = f^{2} = c$
centre (±a,±a), radius a
Passes through origin
$c = 0$
Parametric point
$(α + r cos θ, β + r sin θ)$
0<=θ<2π
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
$S_{11} > 0 outside, = 0 on, < 0 inside$
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
$S_{1} \equiv x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$
Tangent at (x1,y1) on x^2+y^2=a^2
$x x_{1} + y y_{1} = a^{2}$
Tangent at parameter θ
$x cos θ + y sin θ = r$
on x^2+y^2=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
Line lx+my+n=0 tangent to x^2+y^2=r^2
$n^{2} = r^{2} (l^{2} + m^{2})$
Normal to x^2+y^2=r^2
$x y_{1} - y x_{1} = 0$
passes through centre
Position of a line (d vs r)
$d > r no cut, d = r tangent, d < r two points$
Chord of contact from P
$S_{1} = 0$
P external
Polar of P
$S_{1} = 0$
polar line of pole P
Conjugate points
$S_{12} = 0$
Director circle
$x^{2} + y^{2} = 2 r^{2}$
locus of ⟂ tangents to x^2+y^2=r^2
Angle between tangents from P
$tan \frac{θ}{2} = \frac{r}{S _{11}}$
Radical axis
$S - S_{1} = 0$
locus of equal power w.r.t. two circles
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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