SOLUTION: Find an equation of the ellipse that has center (-2,5) , a minor axis of length 10 , and a vertex at (9,5) .

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Find an equation of the ellipse that has center (-2,5) , a minor axis of length 10 , and a vertex at (9,5) .       Log On


   

Question 1137157: Find an equation of the ellipse that has center (-2,5)
, a minor axis of length 10
, and a vertex at (9,5)
.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Find an equation of the ellipse that has

-a center (-2,5)->h=-2, k=5
-a minor axis of length+10->2b=10->b=5
-and a vertex at (9,5)

%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1.....plug in h=-2,+k=5, and b=5
%28x-%28-2%29%29%5E2%2Fa%5E2%2B%28y-5%29%5E2%2F5%5E2=1
%28x%2B2%29%5E2%2Fa%5E2%2B%28y-5%29%5E2%2F25=1

a vertex at (9,5), use it to find a%5E2

%289%2B2%29%5E2%2Fa%5E2%2B%285-5%29%5E2%2F25=1
11%5E2%2Fa%5E2%2B0%2F25=1
121=a%5E2

%28x%2B2%29%5E2%2F121%2B%28y-5%29%5E2%2F25=1



Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.

            To solve this problem,  the correct logical chain of arguments should be build  (and presented)  in a right way.

            I do not see this correct logical chain in the solution of the other tutor,  so I came to present here this logical chain

            and the solution  as it should be. 


(1)  Since the center and the vertex have the same y-coordinates  y= 5, it means that the corresponding semi-axis is horizontal 

     and has the length of  9 - (-2) = 9 + 2 = 11 units.

     Thus we have horizontal semi-axis of 11 units long.



(2)  Now, the minor semi-axis is  10%2F2 = 5 units long, as it follows from the given part.

     Since this length is different from 11 units, it means that the minor semi-axis is vertical, parallel to y-axis.


     It also means that 11-unit semi-axis is the major-semi-axis.

     

(3)  Now we have the full information, geometrically describing the given ellipse and, hence, we are ready to write the equation


         %28x-%28-2%29%29%5E2%2F11%5E2 + %28y-5%29%5E2%2F5%5E2 = 1,    or,  equivalently,


         %28x%2B2%29%5E2%2F11%5E2 + %28y-5%29%5E2%2F5%5E2 = 1.


     It is your final answer.

Solved.

Now you have not only right equation, but the full and correct logical chain of arguments which leads to the equation.