Precalculus : Determine the Equation of a Hyperbola in Standard Form

Example Questions

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Example Question #1 : Determine The Equation Of A Hyperbola In Standard Form

Express the following equation for a hyperbola in standard form:

Explanation:

Remember that in order for the equation of a hyperbola to be in standard form, it must be written in one of the following two ways:

Where the point (h,k) gives the center of the hyperbola. In the first option, where the x term is in front of the y term, the hyperbola opens left and right. In the second option, where the y term is in front of the x term, the hyperbola opens up and down. In either case, the distance tells how far above and below or to the left and right of the center the vertices of the hyperbola are. We can see that our equation is of the form given in the second option, as the y term appears first. All we must do to put it in standard form is obtain a 1 on the right side, so we'll divide both sides by 144:

Now we just simplify the fractions on the left side, and the equation of the hyperbola is in standard form:

Example Question #2 : Determine The Equation Of A Hyperbola In Standard Form

Which of the following is an equation for a hyperbola written in standard form?

Explanation:

In order for the equation of a hyperbola to be in standard form, it must be written in one of the following two ways:

Where the point (h,k) gives the center of the hyperbola, a is half the length of the axis for which it is the denominator, and b is half the length of the axis for which it is the denominator. The easiest requirements of standard form to identify are that the right side of the equation must be 1, there must be subtraction of the terms on the left side and not addition, and the denominators must be different for the x and y terms. Looking at our answer choices, we can see that the following equation is the only one which satisfies these requirements for the standard form of a hyperbola:

Example Question #3 : Determine The Equation Of A Hyperbola In Standard Form

Find the standard equation for a hyperbola defined by the following characteristics:

Vertices:

Foci:

Explanation:

Due to the fact that the foci and vertices each share the same x-coordinate, this particle hyperbola can be classified as a vertical transverse axis hyperbola. Thus, it has an equation of the form:

The point (h,k) is the coordinate of the center point, which is the midpoint of the foci. The values a and c are the distances of the midpoint from the vertices and foci, respectively. The b value can be found from a and c via the Pythagorean Theorem.

Putting it all together, the formula for the hyperbola is:

Example Question #4 : Determine The Equation Of A Hyperbola In Standard Form

Write the equation for a hyperbola in standard form given by the equation:

Explanation:

First, each term should be grouped by like terms (i.e. terms consisting of x's and y's) before factoring. 25 and 16 can be factored out of the x and y groups, respectively, leading to an equation of the form:

Then, one must "complete the squares", or add a term to each group to make them factorable into the form . You should get something that looks like this:

After adding everything together on the right side, condensing the factored out form of the x/y terms, and dividing through by their coefficients, we get the standard form equation:

Example Question #5 : Determine The Equation Of A Hyperbola In Standard Form

Find the standard equation for a hyperbola defined by the following characteristics:

Vertices:

Foci:

Explanation:

Since the x-coordinates of each foci are the same, this hyperbola is transverse axis vertical. The center is the midpoint of the foci, which is  in this case. The a and c values are found by calculating the distance between the center and vertices and foci, respectively.

In this problem,  and . The b value is found through the Pythagoreom Theorem, .

Plugging the values for the center  and a, b, and c, the following equation can be used to find the standard form of the parabola:

Example Question #6 : Determine The Equation Of A Hyperbola In Standard Form

Write the equation for in standard form.

Explanation:

To put this into standard form, we will have to complete the square for both x and y. To do this, it will be really helpful to re-group our terms:

Factor the -2 out of the y terms:

Adding 4 completes the square for x, since

Adding 9 completes the square for y, since

We're adding this inside the parentheses, so really we are adding to both sides

re-write the left side with squared binomials, and add the numbers on the right side

divide both sides by 17

Example Question #7 : Determine The Equation Of A Hyperbola In Standard Form

Write the equation for in standard form

Explanation:

To write this in standard form, first we will have to complete the square for both x and y. It will be helpful to re-group the terms:

Factor out the coefficients for both x squared and y squared

Adding 1 completes the square for x since

Adding 16 completes the square for y since

divide both sides by 6

Example Question #8 : Determine The Equation Of A Hyperbola In Standard Form

Write the equation for in standard form

Explanation:

To write this in standard form, first complete the square for x and y. It will be helpful to re-group our terms:

factor out the coefficients on the squared terms

Adding 9 will complete the square for x since

We're adding it in the parentheses, so really we're adding to both sides

Adding 1 will complete the square for y since

We're really adding to both sides

divide both sides by 36

Example Question #9 : Determine The Equation Of A Hyperbola In Standard Form

Suppose a hyperbola opens up or down.  The center is .  Which of the following is a possible equation in standard form?

Explanation:

The given clue is that the hyperbola will open either up or down.  This indicates that the regular  and  variables are swapped.

If the hyperbola opens left or right, the equation of the hyperbola is:

Write the form of a hyperbola in the second form.

Substitute the center .

The hyperbola must minimally be in this form.

Example Question #10 : Determine The Equation Of A Hyperbola In Standard Form

Write the equation in standard form.

Explanation:

Re-write the equation so that the x terms are next to each other:

factor our 4 from the x terms, and add 20 to both sides

complete the square for x by adding 1 inside the parentheses. To keep both sides equal, add 4 to the right side as well, since the 4 is really being distributed to the 1 we added on the left.

re-write the x terms as a binomial squared

divide both sides by 24

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