Circles

Circle with centre (h, k) and radius r. When centre is the origin: x² + y² = r².

General second-degree equation representing a circle. Coefficients of x² and y² must be equal and non-zero; no xy term.

Centre of x² + y² + 2gx + 2fy + c = 0. Read off as the negatives of half the linear coefficients.

Radius of x² + y² + 2gx + 2fy + c = 0. Real circle requires g² + f² − c > 0; point circle if = 0; imaginary if < 0.

Circle with (x₁, y₁) and (x₂, y₂) as endpoints of a diameter. Follows directly from the angle-in-semicircle being 90°.

Any point on the circle (x−h)² + (y−k)² = r² in terms of the parameter θ ∈ [0°, 360°). For unit circle centred at origin: (cos θ, sin θ).

For circle S = 0 and point P(x₁, y₁): S₁ > 0 ⟹ P outside; S₁ = 0 ⟹ P on the circle; S₁ < 0 ⟹ P inside.

Line y = mx + c is tangent to x² + y² = a² iff c² = a²(1 + m²), equivalently the perpendicular distance from centre equals radius.

Length of the chord at perpendicular distance d from the centre of a circle of radius r. Maximum (diameter) when d = 0.

Equation of the tangent to x² + y² + 2gx + 2fy + c = 0 at the point (x₁, y₁) lying on it. Obtained by the T = 0 substitution rule.

Tangents to x² + y² = a² with slope m. Two tangents exist for every finite m; they are parallel and symmetric about the centre.

Length of the tangent drawn from external point (x₁, y₁) to the circle S = 0. Valid only when S₁ > 0 (point outside the circle).

Combined equation of the two tangents drawn from external point (x₁, y₁) to circle S = 0, where T = xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c.

Normal to x² + y² + 2gx + 2fy + c = 0 at (x₁, y₁). Every normal to a circle passes through its centre (−g, −f).

Equation of the chord joining the two points of tangency when tangents are drawn from external point (x₁, y₁) to circle S = 0. Same form as the tangent at a point — context distinguishes the two.

Equation of the chord of circle S = 0 whose midpoint is (x₁, y₁). Explicitly: xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = x₁² + y₁² + 2gx₁ + 2fy₁ + c.

Two circles touch externally iff the distance between centres equals the sum of radii. Exactly 3 common tangents in this case.

Two circles touch internally iff the distance between centres equals the absolute difference of radii. Exactly 1 common tangent.

Number of common tangents as a function of the distance d between centres relative to radii r₁ and r₂.

Locus of points having equal tangent lengths to both circles. Always a straight line perpendicular to the line joining the centres. For intersecting circles, it is the common chord.

The point equidistant (in tangent length) from three circles. Found by solving any two of the three radical axis equations simultaneously.

For λ ≠ −1, this represents a circle passing through the two intersection points of S₁ = 0 and S₂ = 0. When λ = −1 it degenerates to the radical axis.

Circle passing through the two points where circle S = 0 meets line L = 0. λ is chosen to impose one additional condition (e.g., passes through a given point).

Angle θ between two circles at a point of intersection, defined as the angle between their tangents (equivalently, their radii) at that point.

Two circles x²+y²+2g₁x+2f₁y+c₁=0 and x²+y²+2g₂x+2f₂y+c₂=0 intersect orthogonally iff this condition holds. Equivalently d² = r₁²+r₂².

Polar of the point (x₁, y₁) with respect to x²+y²=a². If the point lies on the circle, the polar is the tangent at that point. If outside, the polar is the chord of contact.

The point whose polar is the given line. Pole and polar are reciprocal: if Q lies on the polar of P, then P lies on the polar of Q (La Hire's theorem).

Locus of the point from which the two tangents to x²+y²=a² are perpendicular. A concentric circle with radius a√2. For circle (x−h)²+(y−k)²=r², the director circle is (x−h)²+(y−k)²=2r².