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Chapter 6: Problem 25

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

Short Answer

Expert verified
The center of the hyperbola is (1,-2); the vertices are (3,-2) and (-1,-2); the foci are \((1+\sqrt{5}, -2)\) and \((1-\sqrt{5}, -2)\). The equations of the asymptotes are \(y= \frac{1}{2}x-2.5\) and \(y=-\frac{1}{2}x-0.5\) respectively.

Step by step solution

01

Find the center

The center of the hyperbola is given by the coordinates \((h,k)\), where \(h\) and \(k\) are the values that make each binomial expression zero. Looking at the equation, we can tell that \((h,k)=(1,-2)\). Thus, the center of the hyperbola is at the point (1,-2).
02

Find the vertices and foci

The denominators 4 and 1 are the square of the semi-major axis \(a^2\) and the semi-minor axis \(b^2\), respectively. Therefore, \(a=\sqrt{4}=2\) and \(b=\sqrt{1}=1\). The vertices are \((h卤a,k)=(1卤2, -2)\) giving us the points (3,-2) and (-1,-2). The distance from the center to the foci, \(c\), is found using the formula \(c=\sqrt{a^2+b^2}\), which gives us \(c=\sqrt{4+1}=\sqrt{5}\). Therefore, the foci are \((h卤c, k)= (1卤\sqrt{5}, -2)\).
03

Determine the equations of the asymptotes

The equations of the asymptotes are of the form \(y = 卤 \frac{b}{a} (x - h) + k\), where \(h, k\) are the coordinates of the center found in step 1. Substituting \(h=1, k=-2, a=2\), and \(b=1\), the equations of the asymptotes are \(y = 卤 \frac{1}{2} (x - 1) - 2\). Thus, the equations of the asymptotes are \(y= \frac{1}{2}x-2.5\) and \(y=-\frac{1}{2}x-0.5\).
04

Sketch the hyperbola

First, plot the center, vertices, and foci on a graph. Then sketch the opening of the hyperbola along the x-axis because the positive term in the equation involves \(x\) and the equations of the asymptotes. The asymptotes provide guides on how the hyperbola will open and curve.

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