Question 770374
<pre>
I'm trying to read your mind.  

I will guess that the red line is the semi-minor-axis, b.

I will guess that the blue line you want to find connects the top
(co-vertex) to a focal point (or focus).  

The semi-major axis is {{{1/2}}} of the major axis, a, so a = {{{11/2}}}.

{{{drawing(400,1000/3,-6,6,-5,5, arc(0,0,11,8),
line(-5.5,0,5.5,0),red(line(0,0,0,4)),blue(line(0,4,3.77,0)),
locate(1.7,0,c),locate(.14,2,b=4), circle(3.77,0,.1), circle(-3.77,0,.1), 
locate(2,2.3,"?")
  )}}}


First we find c, which is the distance from the center to the focus.

All elipses have the property

c² = a² - b²

c² = {{{(11/2)^2}}} - 4²

c² = {{{121/4}}} - 16

c² = {{{121/4}}} - {{{64/4}}}

c² = {{{57/4}}}

By the Pythagorean theorem,


(blue line)² = c² + b²

(blue line)² = {{{57/4}}} + 4²

(blue line)² = {{{57/4}}} + {{{16}}}

(blue line)² = {{{57/4}}} + {{{64/4}}}

(blue line)² = {{{(57+64)/4}}}

(blue line)² = {{{121/4}}}

blue line = {{{sqtr(121/4)}}}

blue line = {{{11/2}}}

So the blue line is equal to the semi-major axis.

Edwin</pre>