We need to find the distance between the two points where the green line
intersects the red ellipse.

First find the equation of the green line:

It passes through the origin (0,0) and the point (a,b)

We find its slope:

m = 
 
m =  = 

Using the point-slope formula:

y - y1 = m(x - x1)

y - 0 = (x - 0)

y = x

To find the endpoints of the chord, where
the green line intersects the red ellipse,
we solve the system of equations:



We clear each of fractions:

b²x² + a²y² = a²b²
         ay = bx

Square both sides of the second equation:

       a²y² = b²x²

Substitute b²x² for a²y² in 

b²x² + b²x² = a²b²

      2b²x² = a²b²

         x² = 

         x² = 

          x =  

          x = 

          x =  

Substitute a²y² for b²x² in 

a²y² + a²y² = a²b²

      2a²y² = a²b²

         y² = 

         y² = 

          y =  

          y = 

          y = 

So the end points of the chord are

 and  


We use the distance formula to find the length of the chord:

d = 

d = 

d = 

d = 

d = 

d = 

That's it.

Edwin