Given the equation of a hyperbola in standard form, locate its vertices and foci.
Determine whether the transverse axis lies on the
x - or
y -axis. Notice that
is always under the variable with the positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the case where the hyperbola is centered at the origin, the intercepts coincide with the vertices.
If the equation has the form
then the transverse axis lies on the
x -axis. The vertices are located at
and the foci are located at
If the equation has the form
then the transverse axis lies on the
y -axis. The vertices are located at
and the foci are located at
Solve for
using the equation
Solve for
using the equation
Locating a hyperbola’s vertices and foci
Identify the vertices and foci of the
hyperbola with equation
The equation has the form
so the transverse axis lies on the
y -axis. The hyperbola is centered at the origin, so the vertices serve as the
y -intercepts of the graph. To find the vertices, set
and solve for
The foci are located at
Solving for
Therefore, the vertices are located at
and the foci are located at
Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.
Hyperbolas centered at the origin
Reviewing the standard forms given for hyperbolas centered at
we see that the vertices, co-vertices, and foci are related by the equation
Note that this equation can also be rewritten as
This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.
Given the vertices and foci of a hyperbola centered at
write its equation in standard form.
Determine whether the transverse axis lies on the
x - or
y -axis.
If the given coordinates of the vertices and foci have the form
and
respectively, then the transverse axis is the
x -axis. Use the standard form
If the given coordinates of the vertices and foci have the form
and
respectively, then the transverse axis is the
y -axis. Use the standard form
Find
using the equation
Substitute the values for
and
into the standard form of the equation determined in Step 1.
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?