Question

The starships of the Solar Federation are marked with the symbol of the Federation, a circle, whereas starships of the Denebian Empire are marked with the Empire's symbol, an ellipse whose major axis is n times its minor axis (a=nb in the figure ).
How fast, relative to an observer, does an Empire ship have to travel for its markings to be confused with those of a Federation ship? Use c for the speed of light in a vacuum.
Express your answer in terms of n and c.

Answer 1

Complete question

The complete question is shown on the first uploaded image  

Answer:

The velocity is  $v = c* \sqrt{1 - \frac{1}{n^2} }$

Explanation:

From the question we are told that

           a = nb

The length of the minor axis  of  the symbol of the Federation, a circle, seen by the observer at velocity v must be equal to the minor axis(b) of the  Empire's symbol, (an ellipse)

Now this length seen by the observer can be mathematically represented as

        $h = t \sqrt{1 - \frac{v^2}{c^2} }$

Here t  is the actual length of the major axis of of the  Empire's symbol, (an ellipse)

So t = a = nb

and  b is the length of the minor axis of the symbol of the Federation, (a circle) when seen by an observer at velocity v which from the question must be the length of the minor axis of the of the  Empire's symbol, (an ellipse)

 i.e    h = b

So

    $b = nb [\sqrt{1 - \frac{v^2}{c^2} } ]$  

     $[\frac{1}{n} ]^2 = 1 - \frac{v^2}{c^2}$

      $v^2 =c^2 [1- \frac{1}{n^2} ]$

       $v^2 =c^2 [\frac{n^2 -1}{n^2} ]$

        $v = c* \sqrt{1 - \frac{1}{n^2} }$