**Topic**

Pluto moves around the Sun in an orbit that is an ellipse with one of the two focal points being the center of the Sun. Given that this ellipse has a semi-major axis \(a \approx 5,{906.10^6}\left( {km} \right)\) and eccentricity \(e \approx 0.249\) (Source: http://vi. wikimedia.org)

Find the smallest (approximate) distance between Pluto and the Sun

**Detailed explanation**

Choose the coordinate system so that the center of the Earth coincides with the focal point \({F_1}\) of the ellipse

Then the ellipse has the equation \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{b^2}}} = 1\) \((0 < b < a)\)

According to the problem, we have: this ellipse has a semi-major axis \(a \approx 5,{906.10^6}\left( {km} \right)\) and eccentricity \(e \approx 0.249\)

Assume Pluto has coordinates \(M\left( {x;y} \right)\)

Then the distance between Pluto and the Sun is: \(M{F_1} = a + ex\)

Since \(x \ge – a\) \(M{F_1} \ge a – ea \approx 5,{906.10^6} – 0.249.5,{906.10^6} = 4,435.406\left( {km} \) right)\)

So the smallest distance between Pluto and the Sun is approximately 4,435,406 km.