Question

use the figure to answer. select the correct choices that complete the sentence below. if the center of the hyperbola is (3,2) and c = 4, then the coordinates of the foci are and (3,4) (3,2) (7,2) (3,6) (4,2)

Explanation:

Step1: Determine the direction of the hyperbola

From the figure, the hyperbola has a vertical transverse axis. For a hyperbola with center \((h,k)\) and vertical transverse axis, the foci are located at \((h,k + c)\) and \((h,k - c)\).

Step2: Substitute the given values

Given \(h = 3\), \(k = 2\), and \(c = 4\).
For the first focus: \(k + c=2 + 4 = 6\), so the coordinate is \((3,6)\).
For the second focus: \(k - c=2- 4=-2\). But looking at the options, we assume there might be a mis - interpretation (if we consider the distance from the center along the vertical axis in a non - standard way, but using the formula for foci of a hyperbola \((h,k\pm c)\) with given center \((3,2)\) and \(c = 4\)).
If we consider the options provided, and re - check the formula application.
The foci of a hyperbola with center \((h,k)\) and distance from center to foci \(c\) (for vertical transverse axis) are \((h,k + c)\) and \((h,k - c)\). Substituting \(h = 3\), \(k = 2\), \(c = 4\) gives \((3,2 + 4)=(3,6)\) and \((3,2-4)=(3,- 2)\). But since \((3,-2)\) is not in the options, and if we assume a wrong - axis consideration (but no, the transverse axis is vertical as per the figure), but if we consider the distance from the center in the wrong direction (which is not correct in hyperbola definition), but among the options \((3,6)\) is present.

Answer:

\((3,6)\) and (assuming a mis - print in options and using the formula \((h,k + c)\) and \((h,k - c)\) with the given center \((3,2)\) and \(c = 4\), if we consider only the positive \(y\) - direction which is wrong in hyperbola terms but as per options) \((3,-2)\) is not there. If we consider a wrong approach (which is not mathematical), but if we use the center \((3,2)\) and \(c = 4\) as distance from center to foci along \(y\) - axis, the foci are \((3,6)\) (since \(2+4 = 6\)) and (if we consider a wrong \(c\) value interpretation, but no) another value. But strictly by formula \((h,k + c)=(3,6)\) and \((h,k - c)=(3,-2)\). But among the given options \((3,6)\) is correct. So the answer is \((3,6)\) (and if we assume the other focus is not in options due to a problem error, but if we have to pick from given options \((3,6)\) is the only one that fits \(h = 3\), \(k + c\) with \(k = 2\), \(c = 4\))