Essential Concepts

Circles

• A circle is all points in a plane that are a fixed distance from a given point on the plane. The given point is called the center, and the fixed distance is called the radius.
• The equation can be written in standard or general form
• The standard form of the equation of a circle with center [latex](h,k)[/latex] and radius [latex]r[/latex] is [latex](x-h)^2+(y-k)^2 = r^2[/latex].
• The general form of a circle is as follows: [latex]x^2+y^2+ax+by+c = 0[/latex].

Ellipses

• An ellipse is the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
• Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci.
• When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form.
• When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse.
• When you start with a more complex form of an ellipse, the key is to transform it into the standard form. This transformation process often involves completing the square and sometimes adjusting the coordinate system.
• When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse.
• When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse.
• When you start with a more complex form of an ellipse, the key is to transform it into the standard form. This transformation process often involves completing the square and sometimes adjusting the coordinate system.
• Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci.

Hyperbolas

• A hyperbola is the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the difference of the distances between [latex]\left(x,y\right)[/latex] and the foci is a positive constant.
• For a hyperbola, the line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
• The standard form of a hyperbola can be used to locate its vertices and foci.
• When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form.
• When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola.
• Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides.

Parabolas

• A parabola is the set of all points [latex]\left(x,y\right)[/latex] in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
• For a parabola, the fixed point is called the focus. The fixed line is called the directrix of the parabola. The axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus. The line segment that passes through the focus and is parallel to the directrix is called the latus rectum. The endpoints of the latus rectum lie on the curve.
• The standard form of a parabola with vertex [latex]\left(0,0\right)[/latex] and the x-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens right. If [latex]p<0[/latex], the parabola opens left.
• The standard form of a parabola with vertex [latex]\left(0,0\right)[/latex] and the y-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens up. If [latex]p<0[/latex], the parabola opens down.
• When given the focus and directrix of a parabola, we can write its equation in standard form.
• The standard form of a parabola with vertex [latex]\left(h,k\right)[/latex] and axis of symmetry parallel to the x-axis can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens right. If [latex]p<0[/latex], the parabola opens left.
• The standard form of a parabola with vertex [latex]\left(h,k\right)[/latex] and axis of symmetry parallel to the y-axis can be used to graph the parabola. If [latex]p>0[/latex], the parabola opens up. If [latex]p<0[/latex], the parabola opens down.
• Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides.

Key Equations

Standard Form	[latex](x-h)^2 + (y-k)^2 = r^2[/latex]
General Form	[latex]x^2 + y^2 + ax + by + c = 0[/latex]
Distance Formula	[latex]d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/latex]
Midpoint Formula	[latex]\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)[/latex]
Line Intersection	[latex]\begin{cases} (x-h)^2 + (y-k)^2 = r^2 \\ y = mx + b \end{cases}[/latex]
Horizontal ellipse, center at origin	[latex]\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center at origin	[latex]\dfrac{{x}^{2}}{{b}^{2}}+\dfrac{{y}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]
Horizontal ellipse, center [latex]\left(h,k\right)[/latex]	[latex]\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center [latex]\left(h,k\right)[/latex]	[latex]\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]
Hyperbola, center at origin, transverse axis on x-axis	[latex]\dfrac{{x}^{2}}{{a}^{2}}-\dfrac{{y}^{2}}{{b}^{2}}=1[/latex]
Hyperbola, center at origin, transverse axis on y-axis	[latex]\dfrac{{y}^{2}}{{a}^{2}}-\dfrac{{x}^{2}}{{b}^{2}}=1[/latex]
Hyperbola, center at [latex]\left(h,k\right)[/latex], transverse axis parallel to x-axis	[latex]\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}-\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex]
Hyperbola, center at [latex]\left(h,k\right)[/latex], transverse axis parallel to y-axis	[latex]\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}-\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}=1[/latex]
Parabola, vertex at origin, axis of symmetry on x-axis	[latex]{y}^{2}=4px[/latex]
Parabola, vertex at origin, axis of symmetry on y-axis	[latex]{x}^{2}=4py[/latex]
Parabola, vertex at [latex]\left(h,k\right)[/latex], axis of symmetry on x-axis	[latex]{\left(y-k\right)}^{2}=4p\left(x-h\right)[/latex]
Parabola, vertex at [latex]\left(h,k\right)[/latex], axis of symmetry on y-axis	[latex]{\left(x-h\right)}^{2}=4p\left(y-k\right)[/latex]

Glossary

center of an Ellipse

The midpoint of both the major and minor axes

center of a Hyperbola

The midpoint of both the transverse and conjugate axes of a hyperbola

conic Section

Any shape resulting from the intersection of a right circular cone with a plane

conjugate Axis

The axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints

directrix

a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant

ellipse

The set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant

focal diameter (latus rectum)

the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola

foci

Plural of focus

focus (of an ellipse)

One of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\left(x,y\right)[/latex] on the ellipse is a constant

focus (of a parabola)

a fixed point in the interior of a parabola that lies on the axis of symmetry

hyperbola

The set of all points [latex]\left(x,y\right)[/latex] in a plane such that the difference of the distances between [latex]\left(x,y\right)[/latex] and the foci is a positive constant

major Axis

The longer of the two axes of an ellipse

minor Axis

The shorter of the two axes of an ellipse

parabola

the set of all points [latex]\left(x,y\right)[/latex] in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix

transverse Axis

The axis of a hyperbola that includes the foci and has the vertices as its endpoints