## Solve Lesson 6 Page 67 Math Study Topic 10 – Kite>

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.