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
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
