A hyperbola has center at the origin, a vertex at (4, 0), and an asymptote given by y = (3/4)x. What is the distance between its foci?
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Correct answer
C. 10
Principle or equation
For a hyperbola centered at the origin with transverse axis along the x-axis, the equation is x^2/a^2 - y^2/b^2 = 1. The asymptotes are y = ±(b/a)x. The distance from the center to each focus is c, where c^2 = a^2 + b^2. The distance between the two foci is 2c.
Why this answer is correct
The vertex is at (4,0), so a = 4. The asymptote y = (3/4)x gives b/a = 3/4, hence b = (3/4)*4 = 3. Then c^2 = a^2 + b^2 = 16 + 9 = 25, so c = 5. The distance between foci is 2c = 10.
Example
If a hyperbola has a = 3 and b = 4, then c = sqrt(9+16)=5, and the distance between foci is 10.
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