A circle has the same center point as an ellipse whose equation is 32x^2+18y^2-128x-180y+290=0.If the circle just fits inside ellipse,what is the diameter of the circle?

Question

anonymous · Answer

Keep in mind a circle is really just an ellipse with major and minor axes being the same length.  An ellipse is really just a stretched out circle in one direction, so we want to find the length of the minor axis of the ellipse and that will be our diameter.  A bit of rearranging...

32x^2 - 128x + 18y^2 - 180y = -290
16x^2 - 64x + 9y^2 - 90y = -145
16(x^2 - 4x + ______) + 9(y^2 - 10y + ______) = -145 + ______ + ______

Complete the square:
16(x^2 - 4x + 4) + 9(y^2 - 10y + 25) = -145 + 64 + 225
16(x - 2)^2 + 9(y - 5)^2 = 144

$$(x - 2)^2/9+(y -5)^2/16 = 1$$

From our ellipse equation, we can see that our a^2 = 9 --> a = 3.  Now, this is only one half of the minor axis, so we only have the radius of the circle.  The diameter is 2*3 = 6.