If a satellite is revolving around a plenet of mass `M` in an elliptical orbit of semi-major axis `a`. Show that the orbital speed of the satellite when it is a distance `r` from the focus will be given by
`upsilon^(2) = GM[(2)/(r ) - (1)/(a)]`

Explanation

To solve the problem, we will use the principles of conservation of energy in the context of a satellite moving in an elliptical orbit around a planet.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a satellite of mass \( m \) revolving around a planet of mass \( M \) in an elliptical orbit. The semi-major axis of the orbit is \( a \), and at a certain point in the orbit, the satellite is at a distance \( r \) from the focus of the ellipse (which is also the center of the planet).

2. **Conservation of Energy**:
- The total mechanical energy \( E \) of the satellite in an elliptical orbit is given by the equation:
\[ E = \text{Kinetic Energy} + \text{Potential Energy} \]
- The total energy \( E \) for an elliptical orbit can also be expressed as:
\[ E = -\frac{GMm}{2a} \]
- Here, \( G \) is the gravitational constant.

3. **Kinetic Energy and Potential Energy**:
- The kinetic energy \( K \) of the satellite at distance \( r \) is:
\[ K = \frac{1}{2} mv^2 \]
- The gravitational potential energy \( U \) at distance \( r \) is:
\[ U = -\frac{GMm}{r} \]

4. **Setting Up the Energy Equation**:
- According to conservation of energy, we can write:
\[ -\frac{GMm}{2a} = \frac{1}{2} mv^2 - \frac{GMm}{r} \]

5. **Rearranging the Equation**:
- We can simplify this equation by multiplying through by \( 2 \) to eliminate the fraction:
\[ -\frac{GMm}{a} = mv^2 - \frac{2GMm}{r} \]
- Rearranging gives:
\[ mv^2 = -\frac{GMm}{a} + \frac{2GMm}{r} \]

6. **Factoring Out \( m \)**:
- Since \( m \) (the mass of the satellite) is common in all terms, we can cancel it out (assuming \( m \neq 0 \)):
\[ v^2 = -\frac{GM}{a} + \frac{2GM}{r} \]

7. **Final Expression**:
- Rearranging the terms gives us:
\[ v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right) \]

### Conclusion:
Thus, we have shown that the orbital speed of the satellite when it is at a distance \( r \) from the focus is given by:
\[ v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right) \]