Learn Conic Sections with Free Lessons & Tips

Why study ellipses?

Orbiting satellites (including the earth revolving around the sun, and the moon revolving around us) trace out elliptical path.

Many buildings and bridges use the ellipse as a pleasing (and strong) shape.

One property of ellipses is that a sound (or any radiation) beginning in one focus of the ellipse will be reflected so it can be heard clearly at the other focus.

Ellipses with Horizontal Major Axis:

Equation for an ellipse with a horizontal major axis is given by:

The ellipse is defined as the locus of a point (x,y) which moves so that the sum of its distances from two fixed points (called foci, or focuses) is constant.

We can produce an ellipse by pinning the ends of a piece of string and keeping a pencil tightly within the boundary of the string, as follows.

We start with these 2 foci:

We pin the ends of the string to the foci and begin to draw, holding the string tight:

Our complete ellipse is formed:

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

How can we obtain a hyperbola from slicing a cone?

We start with a double cone (2 right circular cones placed apex to apex):

When we slice the 2 cones vertically, we get a hyperbola, as shown:                                                                                                                                                                                                        

Cooling towers for a nuclear power plant have a hyperbolic cross-section

How do we create a hyperbola?

Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).

Now, move along a curve such that from any point on the curve,

(distance to A) − (distance to B) = 2a units.

The curve that results is called a hyperbola. There are two parts to the curve.

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

How do we create a hyperbola?

Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).

Now, move along a curve such that from any point on the curve,

(distance to A) − (distance to B) = 2a Units

Cooling towers for a nuclear power plant have a hyperbolic cross-section

General Equation of North-South Hyperbola:

For the hyperbola with focal distance 4a (distance between the 2 foci), and passing through the y-axis at (0, c) and (0, −c), we defineCooling towers for a nuclear power plant have a hyperbolic cross-section

    b2 = c2 − a2

Applying the distance formula for the general case, in a similar fashion to the above example, we obtain the general form for a north-south hyperbola:

    y2/a2-x2/b2 = 1

1. Slope-Intercept form:

Y=mx+b

2. Point-slope form:

The equation of a line passing through a point (x₁, y₁) with slope m

y − y₁ = m(x − x₁)

Genreral Form of Straight line is given by:

Ax+By+C=0

If we slice one of the cones with a plane at right angles to the axis of the cone, the shape formed is a circle.

The circle with centre (0, 0) and radius r has the equation: x² + y² = r²

The circle with centre (h, k) and radius r has the equation: (x − h)2 + (y − k)2 = r2
 

General Form of the Circle: An equation which can be written in the following form (with constants D, E, F) represents a circle: x2 + y2 + Dx + Ey + F = 0x2 + y2 + 2gx + 2fy + c = 0Radius=√g2+f2-ccenter of Circle is given by: -g,-f 
  

The set of all points that share a property. This usually results in a curve or surface.

Example:

Circle:

• A Circle is "the locus of points on a plane that are a certain distance from a central point".
• A circle is the locus of points that are equidistant from a fixed point (the center).

Parabola:

• A parabola is the locus of points that are equidistant from a point (the focus) and a line (the directrix).

Ellipse: