Conic sections

Question

Write an equation in standard form for the ellipse with vertices

(8, 13+2??21) and (8, 13-2??21), and co-vertices (2,13) and (14,13)

 

??=radical sign

thank you!

Answer 1

The first thing to do visualize the ellipse.  Notice that the vertices have the same value of x, and the co-vertices have the same value of y.  that means that the vertices are vertically aligned (above each other), and the co-vertices are horizontally aligned (horizontally "across" from each other).  So the ellipse is vertical, like an egg standing on one end.

 

Next, find the center of the ellipse.  That's going to be the midpoint of the line connecting either the vertices or the co-vertices.  It's easier to do this with the co-vertices, since there are no square roots, but it doesn't matter which.

 

So the midpoint of the line connecting the co-vertices is ((2 + 14) / 2, (13 + 13) / 2) = (16/2, 26/2) = (8, 13).

 

That tells you that the equation is going to look something like this:

 

(x - 8)² / a² + (y - 13)² / b² = 1

 

Then you just have to find a and b.  That's pretty straightforward.  a is equal to the distance from the center to a co-vertex, and b is equal to the distance from the center to a vertex.  (That's because the ellipse is vertically oriented.)

 

To find the distance from the center to a co-vertex, just subtract the x value of a co-vertex from the x value of the center -- so that gives 14 - 8 = 6.

 

Then do the same for b:  The distance from a vertex to the center is 13 + 2√21 - 13 = 2√21.

 

So the equation would be:

 

(x - 8)² / 6² + (y - 13)² / (2√21)² = 1

 

or

 

(x - 8)² / 36 + (y - 13)² / 84 = 1