Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated
units horizontally and
units vertically, the center of the
hyperbola will be
This translation results in the standard form of the equation we saw previously, with
replaced by
and
replaced by
Standard forms of the equation of a hyperbola with center (
h ,
k )
The standard form of the equation of a hyperbola with center
and transverse axis parallel to the
x -axis is
where
the length of the transverse axis is
the coordinates of the vertices are
the length of the conjugate axis is
the coordinates of the co-vertices are
the distance between the foci is
where
the coordinates of the foci are
The asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle is
and its width is
The slopes of the diagonals are
and each diagonal passes through the center
Using the
point-slope formula , it is simple to show that the equations of the asymptotes are
See
[link]a
The standard form of the equation of a hyperbola with center
and transverse axis parallel to the
y -axis is
where
the length of the transverse axis is
the coordinates of the vertices are
the length of the conjugate axis is
the coordinates of the co-vertices are
the distance between the foci is
where
the coordinates of the foci are
Using the reasoning above, the equations of the asymptotes are
See
[link]b .
Like hyperbolas centered at the origin, hyperbolas centered at a point
have vertices, co-vertices, and foci that are related by the equation
We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.
Given the vertices and foci of a hyperbola centered at
write its equation in standard form.
Determine whether the transverse axis is parallel to the
x - or
y -axis.
If the
y -coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the
x -axis. Use the standard form
If the
x -coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the
y -axis. Use the standard form
Identify the center of the hyperbola,
using the midpoint formula and the given coordinates for the vertices.
Find
by solving for the length of the transverse axis,
, which is the distance between the given vertices.
Find
using
and
found in Step 2 along with the given coordinates for the foci.
Solve for
using the equation
Substitute the values for
and
into the standard form of the equation determined in Step 1.
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?