An ellipse with equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) contains the circles \((x - 1)^2 + y^2 = 1\) and \((x + 1)^2 +y^2 = 1.\) Then the smallest possible area of the ellipse can be expressed in the form \(k \pi.\) Find \(k.\)
+128406 The center of the ellipse will be the origin
The length of the major axis is 4....so a is 1/2 of this = 2
The length of the minor axis = 2sqrt(2)....so.....b is 1/2 of this = sqrt(2)
So....the equation of the ellipse is
x^2 y^2
___ + _________ = 1
2^2 [sqrt(2)]^2
x^2 y^2
___ + _____ = 1
4 2
The area of the ellipse = pi * a * b = pi * 2 * sqrt(2) = 2sqrt(2) pi
So
k = 2sqrt (2)
Here is the graph : https://www.desmos.com/calculator/ovso9le6pj
