Standard forms of the equation of an ellipse with center (0,0)
The standard form of the equation of an ellipse with center
and major axis on the
x-axis is
where
the length of the major axis is
the coordinates of the vertices are
the length of the minor axis is
the coordinates of the co-vertices are
the coordinates of the foci are
, where
See
[link]a
The standard form of the equation of an ellipse with center
and major axis on the
y-axis is
where
the length of the major axis is
the coordinates of the vertices are
the length of the minor axis is
the coordinates of the co-vertices are
the coordinates of the foci are
, where
See
[link]b
Note that the vertices, co-vertices, and foci are related by the equation
When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.
(a) Horizontal ellipse with center
(b) Vertical ellipse with center
Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.
Determine whether the major axis lies on the
x - or
y -axis.
If the given coordinates of the vertices and foci have the form
and
respectively, then the major 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 major axis is the
y -axis. Use the standard form
Use the equation
along with the given coordinates of the vertices and foci, to solve for
Substitute the values for
and
into the standard form of the equation determined in Step 1.
Writing the equation of an ellipse centered at the origin in standard form
What is the standard form equation of the ellipse that has vertices
and foci
The foci are on the
x -axis, so the major axis is the
x -axis. Thus, the equation will have the form
The vertices are
so
and
The foci are
so
and
We know that the vertices and foci are related by the equation
Solving for
we have:
Now we need only substitute
and
into the standard form of the equation. The equation of the ellipse is
Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?
Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form
or
Similarly, the coordinates of the foci will always have the form
or
Knowing this, we can use
and
from the given points, along with the equation
to find
Writing equations of ellipses not centered at the origin
Like the graphs of other equations, the graph of an
ellipse can be translated. If an ellipse is translated
units horizontally and
units vertically, the center of the ellipse will be
This
translation results in the standard form of the equation we saw previously, with
replaced by
and
y replaced by
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
