Locus of Midpoint of Tangent to Ellipse

Suppose a tangent is drawn to an ellipse. The ellipse intersects the two axes to form a line of finite length. What is the equation of the midpoint of this line?
A general point on the ellipse is given in parametric coordinates as  
\[(acost, b sint)\]
  and the gradient of the ellipse at this point is  
\[\frac{dy}{dx}= \frac{dy/dt}{dx/dt}=- \frac{bcost}{asint}\]
.
The  
\[y\]
  intercept - using  
\[y=mx+c\]
  - is the solution  
\[c\]
  to  
\[bsint= - \frac{boost}{asint} \times acost+c \rightarrow c=bsint+\frac{bcost}{asint} \times acost=\frac{bsin^2t+bcos^2t}{sint}=\frac{b}{sint}\]

The equation of the tangent is  
\[y=- \frac{bcost}{asint}c+\frac{b}{sint}\]
.
The  
\[x\]
  intercept is the solution to  
\[0=- \frac{bcost}{asint}c+\frac{b}{sint} \rightarrow x= \frac{b/sint}{bcost/asint}=\frac{a}{cost}\]

The  
\[x\]
  intercept is  
\[\frac{b}{sint}\]
.
The coordinates of the intercepts are  
\[(\frac{a}{cost},0)\]
  and  
\[(0,\frac{b}{sint})\]
  and the coordinates of the midpoints is  
\[(\frac{a}{2cost}, \frac{b}{2sint} )\]
.
We get  
\[x=\frac{a}{2cost} \rightarrow cost=\frac{a}{2x}\]
  and  
\[y=\frac{b}{2sint} \rightarrow sint= \frac{b}{2y}\]
.
Then  
\[cos^2+sin^2t=1=\frac{a^2}{4x^2}+\frac{b^2}{4y^2}\]
.

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