+0  
 
0
159
1
avatar

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 \(k.\)

 Apr 5, 2019
 #1
avatar+111400 
0

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

 

 

 

cool cool cool

 Apr 5, 2019

14 Online Users

avatar
avatar
avatar
avatar
avatar