Question

Describe how the equations for an ellipse, circle, hyperbola and parabola differ from one another. Include an example of each in your description.

Answer 1

`\begin{gathered} \text{Ellipse: }\frac{(x^{}-h)^2}{a^2}+^{}((y-k)^2)/(b^2)\text{ = 1} \\ \text{Hyperbola: }\frac{(x^{}-h)^2}{a^2}-^{}((y-k)^2)/(b^2)\text{ = 1} \\ \text{Circle: }(x-h)^2+(y-k)^2=r^2 \\ \text{Parabola: }y=a(x-h)^2+\text{ k} \end{gathered}`

See explanation below

Step-by-step explanation:

An ellipse has a standard equation formula written as:

`\frac{(x^{}-h)^2}{a^2}+^{}((y-k)^2)/(b^2)\text{ = 1}`

When the sign between the x^2 and y^2 terms is positive, then it is a ellipse

`\begin{gathered} \text{example of an ellipse:} \\ ((x-3)^2)/(9)\text{ + }\frac{(y\text{ - 8})^2}{25}\text{ = 1} \end{gathered}`

An hyperbola has a standard equation written as:

`\frac{(x^{}-h)^2}{a^2}-^{}((y-k)^2)/(b^2)\text{ = 1}`

when the sign between the parenthesis of x^2 and y^2 terms is minus, then it is an hyperbola

`\begin{gathered} \text{example of a hyperbola:} \\ (x^2)/(64)\text{ - }\frac{y\text{ }^2}{49}\text{ = 1} \end{gathered}`

A circle has a general formula written as:

`\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ \text{where vertex = (h, k)} \\ r\text{ = radius} \end{gathered}`

The terms x^2 and y^2 are not divided by a constant. Also the left side of the equation represents the square of the radius

`\begin{gathered} An\text{ example of a circle} \\ (x-1)^2+(y-3)^2\text{ = }10 \end{gathered}`

Parabola has a vertex form of equation written as:

`\begin{gathered} y=a(x-h)^2+\text{ k} \\ \text{where a = constant} \end{gathered}`
`\begin{gathered} An\text{ example:} \\ y\text{ = 2(x - 1})^2\text{ + }3 \end{gathered}`

Here, we only have term x^2, no y^2. y has an exponent of 1. Also a constant of a