Problem 27 Find the center, vertices, foci,... [FREE SOLUTION]

Chapter 6: Problem 27

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{(y+6)^{2}}{1 / 9}-\frac{(x-2)^{2}}{1 / 4}=1$$

Short Answer

Expert verified

The center is at (2, -6), vertices are at (2, -5 2/3) and (2, -6 1/3), foci are at (2, -5.40) and (2, -6.60). The equations of the asymptotes are $y = -6 卤 2/3 *(x-2)$.

Step by step solution

Finding the Center

The center of the hyperbola is given by (h, k) in the general formula. Looking at the given equation, it suggests that $h = 2$ and $k = -6$. Therefore, the center of the hyperbola is at the point (2, -6).

Finding the Vertices

The vertices of the hyperbola are points on the hyperbola. In the standard form of the equation, the a value (the square root of the denominator under the $y$ term) determines the distance from the center to the vertices. So here, $a = 1/3$. Since the hyperbola opens upward and downward, we add/subtract a from the y-coordinate of the center to find the vertices. The vertices for this hyperbola are therefore $(2, -6+1/3)$ and $(2, -6-1/3)$, or (2, -5 2/3) and (2, -6 1/3).

Finding the Foci

The square root of the sum of the squares of a and b gives the distance from the center to each focus. In the equation, b is the square root of the denominator under the $x$ term, so $b = 1/2$. Therefore, the distance is $\sqrt{a^2 + b^2} = \sqrt{(1/3)^2 + (1/2)^2} = \sqrt{1/9 + 1/4} = \sqrt{13/36}$ which is $\approx 0.60$. Add and subtract this from the y-coordinate of the center to get the foci, so they are at the points $(2, -6+0.60)$ and $(2, -6-0.60)$, or (2, -5.40) and (2, -6.60).

Finding the equations of the asymptotes

The equations of the asymptotes are given by the formula $y = k 卤 \frac{a}{b}*(x-h)$. So the equations of the asymptotes are $y = -6 卤 2/3 *(x-2)$.

Sketching the Hyperbola

Sketch the anchor points, the center, vertices and foci on a graph. Then, sketch the asymptote lines. These lines give an outline for drawing the hyperbola. Draw the hyperbola arcs touching the asymptotes and passing through the vertices. Be sure to show the direction of the opening of the hyperbola (here, it should open upward and downward).

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91影视

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{(y+6)^{2}}{1 / 9}-\frac{(x-2)^{2}}{1 / 4}=1$$

Short Answer

Step by step solution

Finding the Center

Finding the Vertices

Finding the Foci

Finding the equations of the asymptotes

Sketching the Hyperbola

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