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2021-12-15

What is a hyperbola? What are the Cartesian equations for hyperbolas centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?

Laura Worden

Step 1
Given: To write the definition of the hyperbola:
Definition: In an analytical geometry, a hyperbola is a conic section in which it forms a intersection of the right circular coe with a plane at an angle in such a way that both halves of the cone are intersected.
Therefore, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points (x,y) in the coordinate plane in such a way that the distance between and the focal distance is always a positive constants
Therefore, Cartesian equations for hyperbolas centered at the origin with foci on one of the coordinate axes
$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ (standard equation of hyperbola)
Now, Ellipse is of the general form $\frac{{\left(x-{x}_{1}\right)}^{2}}{{a}^{2}}+\frac{{\left(y-{y}_{1}\right)}^{2}}{{b}^{2}=1}$ with $a>b$
$e=\sqrt{\frac{{a}^{2}-{b}^{2}}{{a}^{2}}}$ is the eccentricity
foci: ; vertices: and equation of directrix is $x=\pi \frac{a}{e}$

Hattie Schaeffer

Step 1
A hyperbola is the of points in a plane whose distamces from two points in the plane have a constant difference.
The Cartesian equation for a hyperbola centered at the origin with foci on the x-axis is given as
$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$
The center-to-focus distance of this hyperbola is
$x=\sqrt{{a}^{2}+{b}^{2}}$
The foci of the hyperbola are the points
The vertices of the hyperbola are the points
The directrices of the hyperbola are given by the equations $x=±c$
The Cartesian equation for a hyperbola centered at the origin with foci on the y-axis is given as
$\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$
The center-to-focus distance of this hyperbola is $c=\sqrt{{a}^{2}+{b}^{2}}$
The foci of th ehyperbola are the points
The vertices of the hyperbola are the points
The directrices of the hyperbola are given by the equations $y=±c$

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