Questions: Find the foci, the eccentricity and then graph (x+6)^2/9 - (y-3)^2/25 = 1. Round your answer to one decimal point if needed.

Transcript text: Find the foci, the eccentricity and then graph $\frac{(x+6)^{2}}{9}-\frac{(y-3)^{2}}{25}=1$. Round your answer to one decimal point if needed.

Solution

Solution Steps

Step 1: Identify the type of conic section

The given equation is: \[ \frac{(x+6)^{2}}{9}-\frac{(y-3)^{2}}{25}=1 \] This is the standard form of a hyperbola.

Step 2: Determine the center, a, and b

The center of the hyperbola is at $(-6, 3)$. The values of $a$ and $b$ are: \[ a^2 = 9 \implies a = 3 \] \[ b^2 = 25 \implies b = 5 \]

Step 3: Calculate the distance to the foci

The distance to the foci $c$ is given by: \[ c = \sqrt{a^2 + b^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.8310 \]

Step 4: Determine the coordinates of the foci

The foci are located at $(h \pm c, k)$: \[ (-6 \pm 5.8310, 3) \] Thus, the foci are approximately: \[ (-0.1690, 3) \quad \text{and} \quad (-11.8310, 3) \]

Step 5: Calculate the eccentricity

The eccentricity $e$ is given by: \[ e = \frac{c}{a} = \frac{5.8310}{3} \approx 1.9437 \]

Final Answer

\[ \text{Foci} = (-0.1690, 3) \quad \text{and} \quad (-11.8310, 3) \] \[ \text{Eccentricity} = 1.9437 \]

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