Question

If y axis is directrix of the ellipse with eccentricity e=1/2 and corresponding focus is at (3,0)then the equation to its auxiliary circle is

Answer 1

Let the equation of the ellipse be

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

The focus of the ellipse is at (3,0).

So, c=3

e=c/a is the equation of Directrix. And the given equation of directrix is e=1/2.

⇒1/2=3/a

⇒a=6,

also, b²=a²- c²

b²=6²- 3²

=36-9

=27

So equation of ellipse is $\frac{x^{2}}{6^{2}} + \frac{y^{2}}{27} =1$

Auxilary circle of an ellipse is the circle passing through two vertices and having centre is equal to center of ellipse.

The vertices are (6,0) and (-6,0).

So the equation of auxiliary circle passing through (6,0) and (-6,0) and center at (0,0) is

$x^{2}  + y ^{2} =6^{2} \\x^{2}  + y ^{2} =36$