maths help

Question

Find an equation for an ellipse with foci (3,4) and (3,6) and vertices (3,1) and (3,9). _________ = 1

                            thanks!!

Answer 1

First we need to define the center that in this case is located at (6-4)/2 = 5=k the center will be at

(3,5). The major axis is parallel to the y axis as we may see for the development of the vertices which move only parallel to the y axis.

Then we have to calculate a, b, c

a is the distance from the center to the vertex along the major axis which becomes (9-1)/2=4

c is the distance from the center to the foci which is (6-4)/2 = 1

we can calculate b which is:  c= √(a²-b²) => 1=(16-b² )=> b²=15

b=√15

Then the equation will be

((x-3)²/15) + ((y-5)/16)= 1

Thanks Ira you are very kind.

 

 

 

 

 

Answer 2

     Sorry in advance to interject, but all francisco's numbers are correct, but if the major axis is in the y direction, the bigger denominator should be under the y term. so you get

 

(x-3)^2/15 +(y-5)^2/16 = 1

     Since you know the vertices, and you know they're on the ellipse, you can substitute them in to check.

 I'll check one of them. one vertex was (3,1). Substituting in you get 0/15 +(-4)^2/16 = 1 which is true.