# Auxiliary Circle of an Ellipse

Auxiliary circle of an ellipse which is a circle described on the major axis of an ellipse as its diameter.

Let the ellipse be $$\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$$ … (1) Then the equation of its auxiliary circle is x² + y² = a² … (2)

Take a point P(x₁, y₁) on (1).

Through P, draw a line perpendicular to major axis intersecting major axis in N and auxiliary circle in P’.

The points P and P’ are called as corresponding points on the ellipse and auxiliary circle respectively.

This angle is known as the eccentric angle of the point P on the ellipse and auxiliary circle respectively.

Examples: Find the equation to the auxiliary circle of the ellipse 4x²+ 9y² – 24x – 36y + 36 = 0.

Solution:

4x²+ 9y² – 24x – 36y + 36 = 0

4 (x² – 6x + 9) + 9 (y² – 4y + 4) = 36

$$\frac{{{\left( x-3 \right)}^{2}}}{9}+\frac{{{\left( y-2 \right)}^{2}}}{4}=1$$.

The ellipse center is (3, 2)

if the length of the major axis of the ellipse be 2a then a² = 9 ⇒ a = 3

x² + y² = a²

x² + y² = 3²