property of ellipse

In my book there is an statement for a ellipse.

Referring to the below ellipse:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

The statement is

The product of the length of the perpendicular segments from the focii on any tangent to the ellipse is $b^2$ and the feet of perpendicular lie on its auxiliary circle and the tangents at these feet to the auxiliary circle meet on the ordinate of P and that the locus of their point of intersection is similar to ellipse as that of the original one.

I could not understand the statement. Can anybody explain it to me with a DIAGRAM(FIGURE) in simple words.

Solution 1:

Hints:

• $\Delta ACD \sim \Delta BCE$
• Let $(\cos \theta,\sin \theta)= \left( \dfrac{AD}{AC},\dfrac{CD}{AC} \right)= \left( \dfrac{BE}{BC},\dfrac{CE}{BC} \right)$ so that $AD\times BE=AC\times BC \cos^2 \theta$
• $AC+BC=2a$
• $AB=2\sqrt{a^2-b^2}$
• $(DC+CE)^2+(AD-BE)^2=AB^2$
• $F$ lies on the ellipse $\dfrac{x^2}{a^2}+\dfrac{b^2y^2}{a^4}=1$

Addendum:

For hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, there are similar properties except the locus gives the re-scaled conjugate hyperbola

$$\frac{x^2}{a^2}-\frac{b^2y^2}{a^4}=1$$

with their transverse axes in common instead of asymptotes.