SOLUTION: Find an equation of the ellipse that has center (1,4) , a minor axis of length 6 , and a vertex at (1,9). answer

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Question 1159590: Find an equation of the ellipse that has center (1,4) , a minor axis of length 6 , and a vertex at (1,9). answer
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Determine whether the major axis is on the x - or y-axis.
If the given coordinates of the vertices and foci have the form (±a,0) and (±c,0) respectively, then the major axis is parallel to the x-axis. Use the standard form:
%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1
If the given coordinates of the vertices and foci have the form (0,±a) and (0,±c) respectively, then the major axis is parallel to the x-axis. Use the standard form:
%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1

given:
a vertex at (1,9)=> is the form (0,±a)
so, go with
%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1

if the center is at (1,4)=>h=1 and k=4

%28x-1%29%5E2%2Fb%5E2%2B%28y-4%29%5E2%2Fa%5E2=1

if the length of minor axis is 6=>2a=6=>a=3

%28x-1%29%5E2%2F3%5E2%2B%28y-4%29%5E2%2Fa%5E2=1

(1,9)=>use to find a
%281-1%29%5E2%2F9%2B%289-4%29%5E2%2Fb%5E2=1
0%2F9%2B25%2Fa%5E2=1
25=a%5E2
your equation is:
%28x-1%29%5E2%2F9%2B%28y-4%29%5E2%2F25=1



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The length of the semi-major axis is the distance from the center to a vertex. The center is (1,4); one vertex is (1,9); the length of the semi-major axis is 5, and it is in the y direction.

Minor axis length 6 means length of semi-minor axis is 3. Since the major axis is in the y direction, the minor axis is in the x direction.

The general equation of an ellipse with center (h,k), semi-major axis a in the y direction and semi-minor axis b in the x direction is

%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2+=+1

Plug in the numbers we were given or have found:

%28x-1%29%5E2%2F3%5E2%2B%28y-4%29%5E2%2F5%5E2+=+1

%28x-1%29%5E2%2F9%2B%28y-4%29%5E2%2F25+=+1