Question

b) \\(\sum_{j=1}^{5} 4^j\\)
graph each ellipse and locate the foci.

1. \\(\frac{(x - 2)^2}{25} + \frac{(y + 1)^2}{9} = 1\\)

Explanation:

Step1: Identify ellipse parameters

Standard form: $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$. Here, $h=2,k=-1,a^2=25,b^2=9$ → $a=5,b=3$.

Step2: Calculate c

Use $c^2=a^2-b^2$ → $c^2=25-9=16$ → $c=4$.

Step3: Find foci coordinates

Foci at $(h\pm c,k)$ → $(2\pm4,-1)$.

Answer:

The foci are $(-2,-1)$ and $(6,-1)$.