Question

Use the description below to find the equation of the hyperbola in standard form. Try making a sketch of the graph to help you. If a value is not an integer type it as a decimal rounded to the nearest hundredth.Standard form:\frac{(x-h)^2}{a^2} -\frac{(y-k)^2}{b^2}=1 Description:Center at (4,2); vertex at (9,2) and one focus at (4+ \sqrt[]{26} ,2) .The value for h is AnswerThe value for k is AnswerThe value for a is AnswerThe value for b is Answer

Answer 1

We have to find the equation of the hyperbola in standard form.

The information we have is:

• Center: (4, 2)

,

• Vertex: (9, 2)

,

• Focus: (4+√26, 2)

From the center, we can derive the values for h and k, as the center of the hyperbola will be (h,k).

Then, h = 4 and k = 2.

We know have to use the information from the vertex and the focus to find parameters a and b.

We can start sketching the hyperbola and the points as:

As 4+√26 is approximately 9.1, the focus and the vertex are very close, as the vertex has x-coordinate x = 9.

We can now calculate the parameter a using the vertex as:

`\begin{gathered} \text{vertex}=(9,2) \\ \Rightarrow((9-4)^2)/(a^2)+((2-2)^2)/(b^2)=1 \\ (5^2)/(a^2)+(0^2)/(b^2)=1 \\ (5^2)/(a^2)=1 \\ a=5 \end{gathered}`

We now have to calculate parameter b.

The only information we haven't used is the focus.

We can relate the focus coordinates to the equation as:

`a^2+b^2=c^2`

where c is the focal distance.

This focal distance will be, in this case, the distance between the focus and the vertex.

We can express this as:

`\begin{gathered} c=\sqrt[]{(x_v-x_f)^2+(y_v-y_f)^2} \\ c=\sqrt[]{(9-4+\sqrt[]{26})^2+(2-2)^2} \\ c=\sqrt[]{(5+\sqrt[]{26})^2} \\ c=5+\sqrt[]{26} \end{gathered}`

We can now calculate b as:

`\begin{gathered} a^2+b^2=c^2 \\ b^2=c^2-a^2 \\ b^2=(5+\sqrt[]{26})^2-5^2 \\ b^2=5^2+2\cdot5\cdot\sqrt[]{26}+(\sqrt[]{26})^2-5^2 \\ b^2=10\sqrt[]{26}+26 \\ b^2\approx77 \\ b\approx\sqrt[]{77} \\ b\approx8.77 \end{gathered}`

Then, we can complete the equation as:

`((x-4)^2)/(5^2)-((y-2)^2)/(8.77^2)=1`

Answer:

h = 4, k = 2, a = 5 and b = 8.77.