ANGELICA P. CASUGBO BTTE II-G Math 212

“HYPERBOLA”

In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola, the ellipse, and the circle; the circle is a special case of the ellipse). If the plane intersects both halves of the double cone but does not pass through the apex of the cones then the conic is a hyperbola.

The hyperbola is centered on a point (h, k), which is the "center" of the hyperbola. The point on each branch closest to the center is that branch's "vertex". The vertices are some fixed distance a from the center. The line going from one vertex, through the center, and ending at the other vertex is called the "transverse" axis. The "foci" of an hyperbola are "inside" each branch, and each focus is located some fixed distance c from the center. (This means that a < c for hyperbolas.) The values of a and c will vary from one hyperbola to another, but they will be fixed values for any given hyperbola.

For any point on an ellipse, the sum of the distances from that point to each of the foci is some fixed value; for any point on an hyperbola, it's the difference of the distances from the two foci that is fixed. Looking at the graph above and letting "the point" be one of the vertices, this fixed distance must be (the distance to the further focus) less (the distance to the nearer focus), or (a + c) – (c – a) = 2a. This fixed-difference property can used for determining locations: If two beacons are placed in known and fixed positions, the difference in the times at which their signals are received by, say, a ship at sea can tell the crew where they are.

As with ellipses, there is a relationship between a, b, and c, and, as with ellipses, the computations are long and painful. So trust me that, for hyperbolas (where a < c), the relationship is c2 – a2 = b2 or, which means the same thing, c2 = b2 + a2. (Yes, the Pythagorean Theorem is used to prove this relationship.

A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone /..hyperbola will be

When the transverse axis is horizontal (in other words, when the center, foci, and vertices line up side by side, parallel to the x-axis), then the a2 goes with the x part of the hyperbola's equation, and the y part is subtracted.

 

   

 

 

When the transverse axis is vertical (in other words, when the center, foci, and vertices line up above and below each other, parallel to the y-axis), then the a2 goes with the y part of the hyperbola's equation, and the x part is subtracted.

   

 

 

A hyperbola is a curve where the distances of any point from:

a fixed point (the focus), and a fixed straight line (the directrix) are always in the same ratio.

This ratio is called the eccentricity, and for a hyperbola it is always greater than 1.

The hyperbola is an open curve (has no ends).

Hyperbola is actually two separate curves in mirror image like this:

On the diagram you can see:

a directrix and a focus (one on each side) an axis of symmetry (that goes through each focus, at right angles to the directrix) two vertices (where each curve makes its sharpest turn)

The "asymptotes" (shown on the diagram) are not part of the hyperbola, but show where the curve would go if continued indefinitely in each of the four directions.

And, strictly speaking, there is also another axis of symmetry that reflects the two separate curves of

the hyperbola.

Equation

By placing a hyperbola on an x-y graph (centered over the x-axis and y-axis), the equation of the curve is:

x2/a2 − y2/b2 = 1

 

Also:

One vertex is at (a, 0), and the other is at (−a, 0)

The asymptotes are the straight lines:

y = (b/a)x y = −(b/a)x

And the equation is also similar to the equation of the ellipse: x2/a2 + y2/b2 = 1, except for a "−" instead of a "+")

Eccentricity

The eccentricity shows how "uncurvy" (varying from being a circle) the hyperbola is. The measure of the amount of curvature is the "eccentricity" e, where e = c/a. Since the foci are further from the center of an hyperbola than are the vertices (so c > a for hyperbolas), then e > 1. Bigger values of e correspond to the "straighter" types of hyperbolas, while values closer to 1 correspond to hyperbolas whose graphs curve quickly away from their centers.

EXAMPLE

On this diagram:

P is a point on the curve, F is the focus and N is the point on the directrix so that PN is perpendicular to the directrix.

The ratio PF/PN is the eccentricity of the hyperbola (for a hyperbola the eccentricity is always greater than 1).

It can also given by the formula:

e = √a2+b2

a

Using "a" and "b" from the diagram above.

Latus Rectum

The Latus Rectum is the line through the focus and parallel to the directrix.

The length of the Latus Rectum is 2b2/a.

 f