Questions: What is (are) the value(s) of x at the focus (foci) of the shape defined by the equation ((x+2)^2)/9 - ((y-1)^2)/16 = 1 ? (Separate multiple values of x with a comma.)

Transcript text: What is (are) the value(s) of $x$ at the focus (foci) of the shape defined by the equation $\frac{(x+2)^{2}}{9}-\frac{(y-1)^{2}}{16}=1$ ? (Separate multiple values of $x$ with a comma.)

Solution

Solution Steps

Step 1: Identify the type of conic section

Given the equation has a minus sign between the fractions, it represents a hyperbola.

Step 2: Calculate the distance $c$ from the center to the foci

Since $a = 3$ and $b = 4$, we calculate $c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = 5$.

Step 3: Calculate the $x$-coordinates of the foci

The foci are located at $(h \pm c, k) = (-2 \pm 5, 1)$, which gives us two points: $(-7, 1)$ and $(3, 1)$.

Final Answer:

The $x$-coordinates of the foci are -7 and 3.

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