Chapter4: Rational Functions And Conics
Section4.3: Conics
Problem 99E
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![**2. Identify the conic as a hyperbola, parabola, or ellipse and find the foci:**
(b) \(\frac{(x - 3)^2}{4} - \frac{(y + 1)^2}{16} = 1\)
**Explanation:**
The given equation is of the form:
\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]
This represents a hyperbola. For this hyperbola:
- The center is at \((h, k) = (3, -1)\).
- Here, \(a^2 = 4\) (so \(a = 2\)) and \(b^2 = 16\) (so \(b = 4\)).
**Finding the Foci:**
For hyperbolas of the form \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\), the distance between the center and each focus along the x-axis (horizontal hyperbola) is given by:
\[
c = \sqrt{a^2 + b^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
Thus, the foci are located at:
\[
(3 + 2\sqrt{5}, -1) \quad \text{and} \quad (3 - 2\sqrt{5}, -1)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40a52fa-ca63-40a5-a54d-3f60ff8c6b34%2Fb9a4cf85-c77a-48df-955f-9cdbdfb88d40%2Fx39m79h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**2. Identify the conic as a hyperbola, parabola, or ellipse and find the foci:**
(b) \(\frac{(x - 3)^2}{4} - \frac{(y + 1)^2}{16} = 1\)
**Explanation:**
The given equation is of the form:
\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]
This represents a hyperbola. For this hyperbola:
- The center is at \((h, k) = (3, -1)\).
- Here, \(a^2 = 4\) (so \(a = 2\)) and \(b^2 = 16\) (so \(b = 4\)).
**Finding the Foci:**
For hyperbolas of the form \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\), the distance between the center and each focus along the x-axis (horizontal hyperbola) is given by:
\[
c = \sqrt{a^2 + b^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
Thus, the foci are located at:
\[
(3 + 2\sqrt{5}, -1) \quad \text{and} \quad (3 - 2\sqrt{5}, -1)
\]
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