When an
ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,
we use the standard forms
for horizontal ellipses and
for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.
Given the standard form of an equation for an ellipse centered at
sketch the graph.
Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
If the equation is in the form
where
then
the center is
the major axis is parallel to the
x -axis
the coordinates of the vertices are
the coordinates of the co-vertices are
the coordinates of the foci are
If the equation is in the form
where
then
the center is
the major axis is parallel to the
y -axis
the coordinates of the vertices are
the coordinates of the co-vertices are
the coordinates of the foci are
Solve for
using the equation
Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graphing an ellipse centered at (
h ,
k )
Graph the ellipse given by the equation,
Identify and label the center, vertices, co-vertices, and foci.
First, we determine the position of the major axis. Because
the major axis is parallel to the
y -axis. Therefore, the equation is in the form
where
and
It follows that:
the center of the ellipse is
the coordinates of the vertices are
or
and
the coordinates of the co-vertices are
or
and
the coordinates of the foci are
where
Solving for
we have:
Therefore, the coordinates of the foci are
and
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.
Given the general form of an equation for an ellipse centered at (
h ,
k ), express the equation in standard form.
Recognize that an ellipse described by an equation in the form
is in general form.
Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
Factor out the coefficients of the
and
terms in preparation for completing the square.
Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant,
where
and
are constants.
Divide both sides of the equation by the constant term to express the equation in standard form.
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?
