Need a little help understanding equation for an ellipse

[nickadams]

In the equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$


can someone explain what those a and b are doing? I know they are the x and y intercepts of the graph, but why are we dividing x^2 and y^2 by them? Also, why are they squared? Why not just regular "a" and "b" like in the parabola equation?


The circle equation makes sense from the pythagorean theorem, but the a and b in the parabola, ellipse, and hyperbola equations really throw me off. Where did they come from? How did people know that if they want to get the U shape of a parabola or C shape of hyperbola or 0 shape of a ellipse that they should make these equations? The circle one is understandable to me since the set of every point that is a certain distance from a point will make a circle... I know my question doesn't make any sense but I can't seem to put into words what I am struggling with...


please help an idiot out!



Edit: why did the equation stack up vertically?

 

[Mark44]

In the equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$


can someone explain what those a and b are doing? I know they are the x and y intercepts of the graph, but why are we dividing x^2 and y^2 by them? Also, why are they squared? Why not just regular "a" and "b" like in the parabola equation?

The numbers a and b represent the major and minor semiaxes of the ellipse, not necessarily in respective order. The major semiaxis is half of the long dimension in the ellipse, and the minor semiaxis is half of the smaller dimension. As such, a and b control the shape of the ellipse.

For example, in the equation
$$\frac{x^2}{4} + \frac{y^2}{3} = 1$$
the major semiaxis is 2 and the minor semiaxis is √3.

From this it can be seen that there are vertices at (2, 0) and (-2, 0), and also at (0, √3) and (0, -√3).

As far as why the formula uses a² and b², it's probably for convenience. If the formula were x²/a + y²/b = 1, it would be difficult to tell if this were an ellipse or a hyperbola without knowing the values of a and b. Having these numbers squared guarantees that the denominators won't be negative.

The circle equation makes sense from the pythagorean theorem, but the a and b in the parabola, ellipse, and hyperbola equations really throw me off. Where did they come from? How did people know that if they want to get the U shape of a parabola or C shape of hyperbola or 0 shape of a ellipse that they should make these equations? The circle one is understandable to me since the set of every point that is a certain distance from a point will make a circle... I know my question doesn't make any sense but I can't seem to put into words what I am struggling with...

The formulas can be derived from the definitions of these conic sections. An ellipse is defined as all of the points that are essentially a fixed distance from two points (the foci of the ellipse).

If you put two pins in a piece of paper taped to some surface the pins can stick in, and tie a piece of string that is longer than the distance between the two pins, you can trace out an ellipse by pulling the string taut with the pencil, and tracing the figure. More formally, if F₁ and F₂ are the two foci of the ellipse, and P(x, y) is an arbitrary point on the ellipse, then F₁P + PF₂ = C, a constant.

A hyperbola has a similar definition.

please help an idiot out!



Edit: why did the equation stack up vertically?

I fixed it. The reason for the odd display was that you used several pairs of tex and /tex tags, and each pair of tags renders on its own line.

 

[nickadams]

Thanks so much for your response, Mark44

As far as why the formula uses a² and b², it's probably for convenience. If the formula were x²/a + y²/b = 1, it would be difficult to tell if this were an ellipse or a hyperbola without knowing the values of a and b. Having these numbers squared guarantees that the denominators won't be negative.

oh okay, that makes sense.

The formulas can be derived from the definitions of these conic sections. An ellipse is defined as all of the points that are essentially a fixed distance from two points (the foci of the ellipse).

If you put two pins in a piece of paper taped to some surface the pins can stick in, and tie a piece of string that is longer than the distance between the two pins, you can trace out an ellipse by pulling the string taut with the pencil, and tracing the figure. More formally, if F₁ and F₂ are the two foci of the ellipse, and P(x, y) is an arbitrary point on the ellipse, then F₁P + PF₂ = C, a constant.

So if you have two foci called "a" and "b" with coordinates of (a,b) and (c,d) respectively, would the equation:

$$\sqrt{(x-a)^2 + (y-b)^2} + \sqrt{(x-c)^2 + (y-d)^2} = C$$

... be an ellipse?

 

[Mark44]

So if you have two foci called "a" and "b" with coordinates of (a,b) and (c,d) respectively,

The foci are traditionally called F₁ and F₂. You could call them a and b, but that wouldn't be very smart when the coordinates of a are a and b, and the coordinates of b are c and d.

would the equation:

$$\sqrt{(x-a)^2 + (y-b)^2} + \sqrt{(x-c)^2 + (y-d)^2} = C$$

... be an ellipse?

It would be the equation of an ellipse, but it wouldn't be an ellipse, if you get my distinction.

 

[nickadams]

The foci are traditionally called F₁ and F₂. You could call them a and b, but that wouldn't be very smart when the coordinates of a are a and b, and the coordinates of b are c and d.

that's true.

It would be the equation of an ellipse, but it wouldn't be an ellipse, if you get my distinction.

yeah, bad wording on my part.

The formulas can be derived from the definitions of these conic sections. An ellipse is defined as all of the points that are essentially a fixed distance from two points (the foci of the ellipse).

Can you go into more detail on these points? How does an ellipse being defined as all points a fixed distance from two points lead to the equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$? How do you go from the definitions of the conic sections (parabola, hyperbola etc.) to their respective equations?

Thanks again

 

[Mark44]

Some of the older calculus books derive the formulas of the conic sections (circle, parabola, ellipse, hyperbola) from the basic definitions. For example, my first calculus book was "Calculus with Analytic Geometry, a First Course," by Murray Protter and Charles Morrey.

For these derivations, there are usually some assumptions to make it simpler to derive the formula. For example, for an ellipse, it is assumed that the center is at (0, 0) and that the two foci are on the horizontal axis. Similar assumptions are made for a circle and a hyperbola.

Deriving the formulas takes some work, so I would suggest that you see what you can find in the way of a calculus text that covers these topics. If it has "analytic geometry" in its name, the chances are good that the text will have these derivations.

I don't know if you will be able to find the book whose name I gave, as it might be out of print. My copy was published in 1963, with the 6th printing in 1967.