- #1
TyroneTheDino
- 46
- 1
Homework Statement
Consider the ellipse ##\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1## that encloses the circle ##x^{2}+y^{2}=2x##. Find the values of a and b that minimize the area of the ellipse.
Homework Equations
##Area=ab\pi##
The Attempt at a Solution
I begin by completing the square of the circle equation to get:
##(x-1)^2+y^2=1##
I note that this circle is centered at (1,0). I know that a>b for a minimal area where the ellipse will touch the circle at 2 points, and if that is so, then a=x of the circle.
I know I need to find a quadratic equation for x in terms of a and b by eliminating y^2. Then derive the ellipse and circle equation implicitly, and set them equal to each other.
After an attempt of using these directions i get:
An ellipse equation of
##\frac{x^{2}}{a^2}+\frac{-(x-1)^2+1}{b^2}=1##
and the circle equation: ##y^2=-(x-1)^2+1##
I hesitate to derive them because I think I'm missing a concept, but if these equations were correct I would derive them and set them equal to each other to find a relation between a and b.
I am not sure if I am on wrong track, but please don't hesitate to tell me if I am misunderstanding a step.