Let P and Q be two points on the ellipse, whose coordinates are (x', y') and (x", y").
The equation to the straight line PQ is
Since both P and Q lie on the ellipse, we have
Hence, by subtraction,
On substituting in (1) the equation to any secant PQ becomes
To obtain the equation to the tangent we take Q indefinitely close to P, and hence, in the limit, we put x" = x' and y" = y'.
The equation (4) then becomes
by equation (2).
The required equation is therefore
