Conic Sections: Hyperbola: Definition and Formula

in #mathematics • 9 years ago (edited)

In this video I go over further into Conic Sections and this time go over the definition of a Hyperbola and derive its resulting formula. I had gone over this derivation several years ago but I have decided to revisit it to be better tie into the Conic Sections video series I have recently been making, as well as to go over it in more detail. The Hyperbola is defined as the set of points on a plane with the property that the difference in the distances from each point to two fixed points (known as the foci) is a constant. Using this definition for the foci on the x-axis, I first show that the difference is equal to +/- 2a because this is the difference when the Hyperbola is also on the x-axis, at either of the two points known as the vertices, and is thus defined as constant for Hyperbolas.

The derivation for the hyperbola is very similar to that for the ellipse, which I covered in my earlier video, and involves using the Pythagorean Theorem for the distances and combined with a lot of algebra to simplify the resulting formulation. After some careful algebra, I show that we can eventually write the hyperbola as x2/a2 – y2/b2 = 1 and where b2 = c2 – a2. The resulting graph of the hyperbola consists of two branches that extend outwards approaching the slant asymptote lines, which I covered in my last two videos, and are y = +/ (b/a)x. This is known as a Horizontal Hyperbola, but we can switch up the x and y terms to get a Vertical Hyperbola y2/a2 – x2/b2 = 1 and with slant asymptotes y = +/- (a/b)x. This is a very extensive video covering the definition and derivation of Hyperbolas so make sure to watch this video!

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Conic Sections: Hyperbolas: Definition and Derivation

Conics Hyperbola Proof AI.jpg

Note: I had made a previous video on the definition of hyperbolas, but I am revisiting it to better tie it to the Conic Section series I am making.

Recall that parabolas, ellipses, and hyperbolas are called conic sections because they result from intersecting a cone with a plane as shown below:

Hyperbolas

A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a constant.

This definition is illustrated below:

Interesting Facts about Hyperbolas

Hyperbolas occur frequently as graphs of equations in chemistry, physics, biology, and economics (Boyle's Law, Ohm's Law, supply and demand curves).

A particularly significant application of hyperbolas is found in the navigation systems developed in World Wars 1 and 2, which I may cover in a later example video so stay tuned!

Notice that the definition of a hyperbola is similar to that of an ellipse; the only change is that the sum of distances has become a difference of distances.

In fact, the derivation of the equation of a hyperbola is also similar to the one given earlier for an ellipse in my earlier video.

Derivation

Let's derive the formula of a hyperbola when the foci are on the x-axis at (+/- c , 0) and the difference of distances is |PF1| - |PF2| = +/- 2a.

Note: The constant +/- 2a can be seen as the vertices of the hyperbola (+/- a , 0) because the distance between the vertices is +/- 2a depending on which distance we subtract from the other one.

Note: If we put x = 0 into the equation we get:

Thus there is no y-intercept, because y2 = - b2 is impossible (unless we deal with imaginary numbers).

The hyperbola is symmetric with respect to both axes:

To analyze the hyperbola further, we notice that:

This means that the hyperbola consists of two parts, called its branches.

When we draw a hyperbola it is useful to first draw its asymptotes, which are the dashed lines y = (b/a)x and y = -(b/a)x, as derived in my earlier videos on Slant Asymptotes.

Both branches of the hyperbola approach the asymptotes; that is, they come arbitrarily close to the asymptotes yet never reach them.

Thus to summarize, the hyperbola is given by the formula:

Similarly, if the foci of a hyperbola are on the y-axis, then by reversing the roles of x and y we obtain the formula for the vertical hyperbola:

#calculus #science #algebra #steemiteducation
9 years ago in #mathematics by
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     9 years ago  

    Math....OMG

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