[tex]\boxed{ \ \frac{(x+4)^2}{4} - \frac{(y-7)^2}{16} = 1 \ }[/tex]
Further explanation
Given:
- The center of a hyperbola is (−4,7)
- One of vertex is (−2,7).
- The slope of one of the asymptotes is 2.
Question:
What is the equation of the hyperbola in standard form?
The Process:
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:
[tex]\boxed{ \ The \ center \ (h, k) \rightarrow The \ vertex \ (h-a, k) \ and \ (h+a, k) \ }[/tex]
Thus, we get the value of a = | -4 - (-2) | = 2.
For a hyperbola with a horizontal transverse axis, the slope of the asymptotes is [tex]\boxed{ \ \pm \frac{b}{a} \ }[/tex].
The slope of one of the asymptotes is 2, therefore we get the value of b, by:
[tex]\boxed{ \ \frac{b}{2} = 2 \rightarrow b = 2 \ }.[/tex]
So we have:
The equation of the hyperbola in standard form with a horizontal transverse axis is
[tex]\boxed{ \ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \ }[/tex]
Let us substitute all the components and get the equation.
[tex]\boxed{\boxed{ \ \frac{(x+4)^2}{4} - \frac{(y-7)^2}{16} = 1 \ }}[/tex]
Notes:
- 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.
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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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