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Coordinate Geometry

Circle

Equations of a circle including standard, center-radius, general, diameter, parametric, tangent, and normal forms.

A circle is the locus of points equidistant from a fixed centre. The fixed distance is the radius r; the centre is the unique point from which every point on the curve is r away. The equations below give the same object in several algebraic forms (standard, centre-radius, general, diameter, parametric), each suited to a different kind of given data. The page also covers tangent and normal lines at a chosen point of contact, in both standard and general settings.

xy(h, k)r
Standard form: centered at (h, k) with radius r.

Standerd Equation

The standard equation places the centre at the origin. Every point (x, y) on the circle satisfies x2 + y2 = a2: the squared distance from the origin equals the squared radius. The parameter a must be positive; a = 0 collapses the circle to a single point and a2 < 0 has no real solutions. Use this form for problems with rotational symmetry about the origin.

x2+y2=a2 (a=radius)x^{2}+y^{2}=a^{2}\text{ }\left(a=radius\right)

Center-Radius Form

Translating the centre to (h, k) shifts each variable by the corresponding centre coordinate. The equation states that the squared distance from (x, y) to (h, k) equals r2. Setting h = k = 0 recovers the standard form. The radius must be non-negative for the equation to describe a real curve.

(x−h)2+(y−k)2=r (r=radius, (h,k)=center)\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{}\text{ }\left(r=radius,\text{ }\left(h,k\right)=center\right)

General Equation of a Circle

Expanding (x − h)2 + (y − k)2 = r2 and renaming constants gives the general quadratic form. Any second-degree equation in x and y with equal coefficients on x2 and y2 and no xy term reduces to this shape. The general form is the natural output of intersection or locus computations, where centre and radius are not yet known.

x2+y2+2gx+2fy+c=0x^{2}+y^{2}+2gx+2fy+c=0

Comparing with the expanded centre-radius form, the centre is read off the linear coefficients as (−g, −f).

Center=(−g,−f)Center = \left(-g,-f\right)

The radius comes from completing the square. The curve is a real circle only when the radicand g2 + f2 − c is positive (a point if zero, imaginary if negative).

Radius=g2+f2−cRadius = \sqrt{g^{2}+f^{2}-c}

Diameter Form

Given two diametrically opposite endpoints (x1, y1) and (x2, y2), the circle is the locus of points P such that ∠(x1P x2) is a right angle (Thales’ theorem). The dot product of the vectors from P to each endpoint is zero, giving the diameter form. The centre is the midpoint of the diameter; the radius is half its length.

(y−y1)(y−y2)+(x−x1)(x−x2)=0\left(y-y_{1}\right)\left(y-y_{2}\right)+\left(x-x_{1}\right)\left(x-x_{2}\right)=0

Tangent at P(x1,y1) of a Standard Circle

The tangent line at (x1, y1) on the circle x2 + y2 = a2 is perpendicular to the radius at that point. Apply the “T = 0” rule: replace x2 by x x1 and y2 by y y1. The contact point must satisfy the original equation; otherwise the resulting line is the polar of an external point, not a tangent.

xx1+yy1=a2xx_{1}+yy_{1}=a^{2}

Normal at P(x1,y1) of a Standard Circle

The normal at any point on a circle passes through that point and the centre. For the standard circle the centre is the origin. The equation y / y1 = x / x1 says that (x, y) is a scalar multiple of (x1, y1), i.e. lies on the line through the origin and the contact point. The form breaks down when either x1 or y1 is zero; in those cases the normal is a coordinate axis.

yy1=xx1\frac{y}{y_{1}}=\frac{x}{x_{1}}

Tangent at P(x1,y1) of a General Circle

For the general-form circle, the T = 0 rule extends to the linear terms: replace x by (x + x1) / 2 and y by (y + y1) / 2, then clear the halves. The contact point must lie on the circle for the result to be a true tangent. Setting g = f = 0 recovers the standard-form tangent.

xx1+yy1+g(x+x1)+f(y+y1)+c=0xx_{1}+yy_{1}+g\left(x+x_{1}\right)+f\left(y+y_{1}\right)+c=0

Normal at P(x1,y1) of a General Circle

The normal at (x1, y1) on the general circle is the line joining the contact point to the centre (−g, −f). The two-point form, with the centre as the second point, yields the equation below. The line is well defined whenever the contact point and centre are distinct, which holds for any point on the circle.

(y−y1x−x1)=(y1−fx1−g)\left(\frac{y-y_{1}}{x-x_{1}}\right)=\left(\frac{y_{1}-f}{x_{1}-g}\right)

Condition For a Line To Be a Tangent

For the line y = mx + c to be tangent to the standard circle x2 + y2 = a2, the perpendicular distance from the origin to the line must equal a. Substituting y = mx + c into the circle equation and demanding a double root gives c2 = a2(m2 + 1). Two values of c (one positive, one negative) satisfy this, giving the two parallel tangents of slope m.

c2=a2(m2+1)c^{2}=a^{2}\left(m^{2}+1\right)

When the tangency condition holds, the line touches the circle at a single point, obtained by solving the substituted equation as a perfect square. The contact point depends on the sign of c, so the two parallel tangents of slope m touch opposite points.

Point of contact=(−a2mc,a2c)Point\text{ }of\text{ }contact = \left(\frac{-a^{2}m}{c},\frac{a^{2}}{c}\right)

Parametic Equation of Standard Circle

A single angular parameter θ traces the standard circle: the angle measured at the centre from the positive x-axis to the radius at (x, y). As θ sweeps from 0 to 2π, the point (a cos θ, a sin θ) traces the circle once counter-clockwise. Use this form for integrating around the circle or parameterising motion along it.

x=a.cos⁡θx=a.\cos\theta
u=a.sin⁡θu=a.\sin\theta

Parametic Equation of General Circle

For a circle centred at (h, k) with radius r, the parametric form adds the centre coordinates as offsets to the standard parameterisation. Each point on the circle is the centre plus a radius-vector of length r at angle θ. In animation and physics θ often plays the role of time.

x=h+r.cos⁡θx=h+r.\cos\theta
x=h+r.sin⁡θx=h+r.\sin\theta
[Center=(h,k) and radius=r]\left[Center=(h,k)\text{ }and\text{ }radius=r\right]

Tangent To a Standard Circle at Point P(a.cosθ, a.sinθ)

Substituting the parametric coordinates into x x1 + y y1 = a2 and dividing by a gives the tangent in angular form. Use this when the contact point is identified by angle rather than Cartesian coordinates. The coefficients (cos θ, sin θ) are the unit normal direction from the centre to the contact point.

y.sin⁡θ+x.cos⁡θ=ay.\sin\theta+x.\cos\theta=a

Normal To a Standard Circle at Point P(a.cosθ, a.sinθ)

The normal at a parametric point is the line through the origin and the contact point, in the direction (cos θ, sin θ). The equation x / cos θ = y / sin θ says (x, y) is proportional to (cos θ, sin θ). The form is undefined when cos θ or sin θ is zero, where the normal coincides with an axis.

xcos⁡θ=ysin⁡θ\frac{x}{\cos\theta}=\frac{y}{\sin\theta}