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Chapter 6: Problem 43

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: \((1,2),(3,2)\) asymptotes: \(y=x, y=4-x\)

Short Answer

Expert verified
The standard form of the hyperbola is \( (x - 2)^2 - (y - 2)^2 = 1 \).

Step by step solution

01

Find the center of the hyperbola

The mid-point between the vertices of a hyperbola is the center of the hyperbola. To find the midpoint of \( (1,2) \) and \( (3,2) \), use the midpoint formula \((x_1+x_2)/2, (y_1+y_2)/2 \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Thus, center \( (h,k) = (2,2) \).
02

Find the lengths 2a and 2b

For determine the value of \( a \), find the distance between one vertex and the center of the hyperbola. Since the center is \( (2,2) \) and one vertex is \( (1,2) \), the distance is \(1\); so \( a = 1 \). The value of \( b \) is found using the formula \( a^2 + b^2 = c^2 \), where the distance c between the asymptote and the center of the hyperbola, can be found from the equation of the asymptote \( y = x \). When \( x = 2 \), \( y = 2 \). So, \( c = 0 \). Solving the above equation for \( a=1 \), \( b=1 \) is found.
03

Write the equation of the hyperbola

Since the center is \( (h,k) = (2,2) \), the value of \( a = 1 \), and \( b = 1 \), and the slopes of the asymptotes indicate that the hyperbola is opened to left and right, the equation of the hyperbola in standard form is \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\). Plugging the values for \( h, k, a, \) and \( b \), we get \( (x - 2)^2 - (y - 2)^2 = 1 \).

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Most popular questions from this chapter

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{2}{1+\sin \theta}$$

Determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=6$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=5 \pi / 3$$
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