The equation that models Pluto's elliptical orbit around the sun is $$ \frac{(x+0.9)^{2}}{12.96}+\frac{y^{2}}{12.15}=1 $$ in billions of miles. If the sun is located at one focus and its coordinates are $$ (0,0) $$, find Pluto's farthest distance from the sun in billions of miles. (1 point) 13.86 billion mi . 12.06 billion mi . 4.50 billion mi. 2.70 billion mi.

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Pluto's farthest distance from the sun is **4.50 billion mi**.

**Explanation:**

The given ellipse equation is $$\frac{(x+0.9)^2}{12.96} + \frac{y^2}{12.15} = 1$$.

- **Center of the ellipse:** $$(-0.9, 0)$$
- **Semi-major axis (a):** $$\sqrt{12.96} = 3.6$$ billion miles
- **Semi-minor axis (b):** $$\sqrt{12.15} \approx 3.485$$ billion miles
- **Distance to the focus (c):** $$\sqrt{a^2 - b^2} = \sqrt{12.96 - 12.15} = 0.9$$ billion miles

Since the sun is at one focus $$(0,0)$$, the farthest point on the ellipse from the sun is at $$(-0.9 - 3.6, 0) = (-4.5, 0)$$. The distance is $$4.5$$ billion miles.

**Answer:** 4.50 billion mi.
Pluto's farthest distance from the sun is **4.50 billion miles**.

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