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