Area of Ellipse/Proof 2

Proof
Let $K$ be an ellipse aligned in a cartesian plane in reduced form.

Then from Equation of Ellipse in Reduced Form:
 * $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

Thus:
 * $y = \pm \dfrac b a \sqrt {a^2 - x^2}$

Consider a circle of radius $a$ whose center is at the origin.

From Equation of Circle center Origin, its equation is given by:
 * $x^2 + y^2 = a^2$

and so:
 * $y = \pm \sqrt {a^2 - x^2}$

The formulas show that each ordinate of the ellipse is $\dfrac b a$ the ordinate of the circle.

Since the same thing is true of the vertical chords: