![\boxed{ \ \frac{(x+4)^2}{4} - \frac{(y-7)^2}{16} = 1 \ }](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5C%20%5Cfrac%7B%28x%2B4%29%5E2%7D%7B4%7D%20-%20%5Cfrac%7B%28y-7%29%5E2%7D%7B16%7D%20%3D%201%20%5C%20%7D)
<h3>
Further explanation</h3>
<u>Given:</u>
- The center of a hyperbola is (−4,7)
- One of vertex is (−2,7).
- The slope of one of the asymptotes is 2.
<u>Question:</u>
What is the equation of the hyperbola in standard form?
<u>The Process:</u>
The center of a hyperbola is (−4,7), we call as (h, k).
One of the vertices is (−2,7), the same ordinate as the center, so we have hyperbola with a horizontal transverse axis.
For a hyperbola with a horizontal transverse axis, the relationship between the center and vertex is as follows:
![\boxed{ \ The \ center \ (h, k) \rightarrow The \ vertex \ (h-a, k) \ and \ (h+a, k) \ }](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5C%20The%20%5C%20center%20%5C%20%28h%2C%20k%29%20%5Crightarrow%20The%20%5C%20vertex%20%5C%20%28h-a%2C%20k%29%20%5C%20and%20%5C%20%28h%2Ba%2C%20k%29%20%5C%20%7D)
Thus, we get the value of a = | -4 - (-2) | = 2.
For a hyperbola with a horizontal transverse axis, the slope of the asymptotes is
.
The slope of one of the asymptotes is 2, therefore we get the value of b, by:
![\boxed{ \ \frac{b}{2} = 2 \rightarrow b = 2 \ }.](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5C%20%5Cfrac%7Bb%7D%7B2%7D%20%3D%202%20%5Crightarrow%20b%20%3D%202%20%5C%20%7D.)
So we have:
The equation of the hyperbola in standard form with a horizontal transverse axis is
![\boxed{ \ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \ }](https://tex.z-dn.net/?f=%5Cboxed%7B%20%5C%20%5Cfrac%7B%28x%20-%20h%29%5E2%7D%7Ba%5E2%7D%20-%20%5Cfrac%7B%28y%20-%20k%29%5E2%7D%7Bb%5E2%7D%20%3D%201%20%5C%20%7D)
Let us substitute all the components and get the equation.
![\boxed{\boxed{ \ \frac{(x+4)^2}{4} - \frac{(y-7)^2}{16} = 1 \ }}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cboxed%7B%20%5C%20%5Cfrac%7B%28x%2B4%29%5E2%7D%7B4%7D%20-%20%5Cfrac%7B%28y-7%29%5E2%7D%7B16%7D%20%3D%201%20%5C%20%7D%7D)
<u>Notes:</u>
- Hyperbola is one part of the conic section. Hyperbola is a set of all points that the difference in distance to two specific points (focus) has a constant value.
- On the transverse axis of the hyperbola are two focus points, two vertices, and one center point.
- The center of a hyperbola is in the middle between the foci.
- If the x²-term in the equation of a hyperbola is positive, the transverse axis lies on the x-axis, i.e., horizontal transverse axis.
- If the y²-term in the equation of a hyperbola is positive, the transverse axis lies on the y-axis, i.e., vertical transverse axis.
<h3>Learn more</h3>
- What is the general form of the equation of the given circle with center A(-3,12) and the radius is 5? brainly.com/question/1506955
- The piecewise-defined functions brainly.com/question/9590016
- Which phrase best describes the translation from the graph y = 2(x – 15)2 + 3 to the graph of y = 2(x – 11)2 + 3? brainly.com/question/1369568
Keywords: the center, a hyperbola, vertex, the slope, the asymptotes, the equation, in standard form, horizontal transverse axis, focus, foci, a plane, the conic section
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