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
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product