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Formulas and Tables

Analytic Geometry

Circle and Ellipse

Radius of a circle: $$R$$
Center of a circle, semi-axes of an ellipse: $$a, b$$
Point coordinates: $$x,$$ $$y,$$ $${x_1},$$ $${y_1}, \ldots$$
Real numbers: $$A,$$ $$B,$$ $$C,$$ $$D,$$ $$E,$$ $$F,$$ $$t$$
Foci of an ellipse: $${F_1}$$, $${F_2}$$

Focal distance: $$2c$$
Eccentricity of an ellipse: $$e$$
Elliptic integral: $$E$$
Circumference: $$L$$
Area: $$S$$

1. A circle is a closed plane curve all points of which are equidistant from a given fixed point called the centre of the circle. The distance from any point of the circle and the centre is called the radius of the circle. The equation of a circle with radius $$R$$ centered at the origin (canonical equation of a circle) has the form
$${x^2} + {y^2} = {R^2}$$.
1. The equation of a circle of radius $$R$$ centered at the point $$A\left( {a,b} \right)$$ is written as
$${\left( {x – a} \right)^2}$$ $$+\; {\left( {y – b} \right)^2}$$ $$= {R^2}$$.
1. The equation of a circle passing through three points (three-point form) is given by
$$\left| {\begin{array}{*{20}{c}} {{x^2} + {y^2}} & x & y & 1\\ {x_1^2 + y_1^2} & {{x_1}} & {{y_1}} & 1\\ {x_2^2 + y_2^2} & {{x_2}} & {{y_2}} & 1\\ {x_3^2 + y_3^2} & {{x_3}} & {{y_3}} & 1 \end{array}} \right|$$ $$= 0.$$
Here $$A\left( {{x_1},{y_1}} \right)$$, $$B\left( {{x_2},{y_2}} \right)$$, $$C\left( {{x_3},{y_3}} \right)$$ are three distinct points lying on the circle.
1. Equation of a circle in parametric form
\left\{ \begin{aligned} x &= R \cos t \\ y &= R\sin t \end{aligned} \right.,\; $$0 \le t \le 2\pi$$,
where $$x$$, $$y$$ are the coordinates of any point of the circle, $$R$$ is the radius of the circle, $$t$$ is a parameter.
2. General form of a circle equation
$$A{x^2} + A{y^2}$$ $$+\; Dx + Ey$$ $$+\; F = 0$$
provided that $$A \ne 0$$, $$D^2 + E^2 \gt 4AF$$.
The centre of the circle has coordinates $$\left( {a,b} \right)$$ where
$$a = – {\large\frac{D}{{2A}}\normalsize},\;$$ $$b = – {\large\frac{E}{{2A}}\normalsize}.$$
The radius of the circle is given by
$$R = \large{\frac{\sqrt{{D^2} + {E^2} – 4AF}}{{2\left| A \right|}}\normalsize}$$
3. An ellipse is a closed plane curve such that the sum of the distances from any point of the curve to two other fixed points (called the foci of the ellipse) is constant. The distance between the foci is called the focal distance and denoted as $$2c$$. The midpoint of the line segment joining the foci is called the center of the ellipse. Any ellipse has two axes of symmetry: the first or focal line passing through the foci and the second line perpendicular to the first axis. The points of intersection of these axes with the ellipse are called the vertices. A line segment that runs from the center of the ellipse to its vertex is called the semi-axis of the ellipse. The semi-major axis is usually denoted by $$a$$, and the semi-minor axis is denoted by $$b$$. An ellipse with the center at the origin and the semi-axes lying on the coordinate lines is described by the following canonical equation:
$${\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 1.$$
1. The sum of the distances from any point of an ellipse to its foci is constant:
$${r_1} + {r_2} = 2a$$,
where $${r_1}$$, $${r_2}$$ are the distances from an arbitrary point $$P\left( {x,y} \right)$$ of the ellipse to the foci $${F_1}$$ and $${F_2}$$, $$a$$ is the semi-major axis of the ellipse.
1. Relationship between the semi-axes of an ellipse and its focal distance
$${a^2} = {b^2} + {c^2}$$,
where $$a$$ is the semi-major axis of the ellipse, $$b$$ is the semi-minor axis, $$c$$ is half of the focal length.
2. Eccentricity of an ellipse
$$e = {\large\frac{c}{a}\normalsize} \lt 1$$
3. Equation of the directrix of an ellipse
The directrix of an ellipse is a straight line perpendicular to the focal axis of the ellipse and intersecting it at the distance $$\large\frac{a}{e}\normalsize$$ from the center. An ellipse has two directrices spaced on opposite sides of the center. The equations of the directrices are written in the form
$$x = \pm {\large\frac{a}{e}\normalsize} = \pm {\large\frac{{{a^2}}}{c}\normalsize}.$$
4. Equation of an ellipse in parametric form
\left\{ \begin{aligned} x &= a\cos t \\ y &= b\sin t \end{aligned} \right.,\; $$0 \le t \le 2\pi$$,
where $$a$$, $$b$$ are the semi-axes of the ellipse, $$t$$ is a parameter.
5. General equation of an ellipse
$$A{x^2} + Bxy + C{y^2}$$ $$+\; Dx + Ey$$ $$+\; F = 0,$$
where $${B^2} – 4AC \lt 0$$.
6. General equation of an ellipse with semi-axes parallel to the coordinate axes
$$A{x^2} + C{y^2}$$ $$+\; Dx + Ey$$ $$+\; F = 0,$$
where $$AC \gt 0$$.
7. Circumference of an ellipse
$$L = 4aE\left( e \right)$$,
where $$a$$ is the semi-major axis, $$e$$ is the eccentricity, $$E$$ is the complete elliptic integral of the second kind.
8. Approximate formulas for the circumference of an ellipse
$$L \approx \pi \left[ {{\large\frac{3}{2}\normalsize}\left( {a + b} \right) – \sqrt {ab} } \right],\;$$ $$L \approx \pi \sqrt {2\left( {{a^2} + {b^2}} \right)},$$
where $$a$$, $$b$$ are the semi-axes of the ellipse.
9. Area of an ellipse
$$S = \pi ab$$