Equations of Hyperbolas Quiz Active 1 2 3 5 6 7 The graph of a hyperbola is shown. (-13, 0) (-12, 0) -21 -18 -15 -12 -9 -6 -3 Mark this and return (0, 5) 5 4 3 2 1 (12, 0) (13, 0) 3 6 9 12 15 18 21 x -1 -2 -3 -4 -5 (0, -5) -6 What are the coordinates of a focus of the hyperbola? (-12, 0) (0, -5) (0, 0) (13, 0) Save and Exit Next Submit
The Correct Answer and Explanation is:
Correct Answer: (13, 0)
Explanation
To find the coordinates of a focus of the hyperbola, we begin by analyzing its graph. From the visual and listed coordinates, we observe the following key points:
- The vertices of the hyperbola are at (-13, 0) and (13, 0).
- The center of the hyperbola is the midpoint of the segment joining the two vertices, which is (0, 0).
- The points (0, 5) and (0, -5) lie on the conjugate axis, indicating the height of the rectangle that guides the asymptotes.
This is a horizontal hyperbola, as the transverse axis (the line passing through the vertices) is horizontal.
The standard form of a horizontal hyperbola centered at (h,k)(h, k) is: (x−h)2a2−(y−k)2b2=1\frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1
Given:
- Center: (h,k)=(0,0)(h, k) = (0, 0)
- Distance from center to each vertex: a=13a = 13
- Distance from center to each co-vertex: b=5b = 5
So the equation becomes: x2132−y252=1⇒x2169−y225=1\frac{x^2}{13^2} – \frac{y^2}{5^2} = 1 \quad \Rightarrow \quad \frac{x^2}{169} – \frac{y^2}{25} = 1
To find the coordinates of the foci, we use the relationship: c2=a2+b2c^2 = a^2 + b^2
Substitute known values: c2=169+25=194⇒c=194≈13.93c^2 = 169 + 25 = 194 \Rightarrow c = \sqrt{194} \approx 13.93
This tells us the foci are approximately 13.93 units left and right of the center along the x-axis.
So the coordinates of the foci are approximately: (±194,0)≈(±13.93,0)(\pm \sqrt{194}, 0) \approx (\pm 13.93, 0)
The point closest to that from the options given is (13, 0).
Hence, the correct answer is (13, 0) — one of the two foci of the hyperbola.
