List of formulas related Analytic geometry (circle)

Equation of a circle:

In an x−y coordinate system, the circle with center (a,b) and radius r is the set of all points (x,y) such that:

$$ (x-a)^2 + (y-b)^2 =r^2 $$

Circle centered at the origin:

$$ x^2 + y^2 = r^2 $$

Parametric equations

$$ \begin{aligned} x &= a + r\,\cos t \\ y&= b + r\,\sin t \end{aligned} $$

where t is a parametric variable.

In polar coordinates the equation of a circle is:

$$ r^2 - 2\cdot r \cdot r_0\cdot cos(\Theta - \phi ) + r_0^2 = a^2 $$

Area of a circle:

$$ A = r^2\pi $$

Circumference of a circle:

$$ C = \pi \cdot d = 2\cdot \pi \cdot r $$


(Chord theorem) The chord theorem states that if two chords, CD and EF, intersect at G, then:

$$ CD \cdot DG = EG \cdot FG $$

Tangent secant theorem

tangent secant Chord theorem

(Tangent-secant theorem) If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then

$$ DC^2 = DG \cdot DE $$

(Secant - secant theorem) If two secants, DG and DE, also cut the circle at H and F respectively, then:

$$ DH \cdot DG = DF \cdot DE $$

(Tangent chord property) The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord.