SOLUTION: Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and focus at (1, -1). a. (9x)2 + (36y)2 - 54x + 50y - 116 = 0 b. (4x)2 + (25y)2 + 54x - 5

Algebra ->  Finance -> SOLUTION: Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and focus at (1, -1). a. (9x)2 + (36y)2 - 54x + 50y - 116 = 0 b. (4x)2 + (25y)2 + 54x - 5      Log On


   

Question 1165474: Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and focus at (1, -1).
a. (9x)2 + (36y)2 - 54x + 50y - 116 = 0
b. (4x)2 + (25y)2 + 54x - 50y - 122 = 0
c. (9x)2 + (25y)2 + 50x + 50y + 109 = 0
d. (9x)2 + (25y)2 + 54x + 50y - 119 = 0
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
the equation of the ellipse is:

%28x-h%29%5E2%2Fa%5E2+%2B+%28y-k%29%5E2%2Fb%5E2+=+1
given:
center is at (-3, -1)=> h=-3, k=-1
so far we have:
%28x%2B3%29%5E2%2Fa%5E2+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1

focus at (1,+-1)
vertex at (2, -1)
The major axis a is the segment that contains both foci and has its endpoints on the ellipse. These endpoints are called the vertices. The midpoint of the major axis is the center of the ellipse.
so using midpoint formula, the other vertex will be at
%282%2Bx%29%2F2=-3=>+2%2Bx=-6=>+x=-6-2=> x=-8
%28-1%2By%29%2F2=-1=>-1%2By=-2=>y=-2%2B1=>y=-1
other vertex will be at (-8,-1)

and the major axis =2a is distance between vertices
2a=sqrt%28%282-%28-8%29%29%5E2%2B%28-1-%28-1%29%29%5E2%29
2a=sqrt%2810%5E2%2B0%5E2%29
2a=10
a=5

so far
%28x%2B3%29%5E2%2F5%5E2+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1
%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1
The formula generally associated with the focus of an ellipse is+c%5E2=a%5E2-b%5E2 where c+is the distance from the focus to center , a+is the distance from the center to a vetex and b+is the distance from the center to a co-vetex

if center is at (-3, -1) and focus at (1, -1)
and c is the distance from the focus to center, then=>
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%28-3-%281%29%29%5E2%29%2B%28%28-1-%28-1%29%29%5E2%29%29
sqrt%28-4%5E2%2B0%5E2%29 The distance is 4.




so, a=5 andc=4
c%5E2=a%5E2-b%5E2
b%5E2=a%5E2-c%5E2
b%5E2=5%5E2-4%5E2
b%5E2=25-16
b%5E2=9
so, your equation is
%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2F9+=+1
expand:
%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2F9+=+1... both sides multiply by 25%2A9=225
225%28x%2B3%29%5E2%2F25+%2B+225%28y%2B1%29%5E2%2F9+=+225..........simplify
9%28x%2B3%29%5E2+%2B+25%28y%2B1%29%5E2+=+225
9%28x%5E2+%2B+6+x+%2B+9%29+%2B+25%28y%5E2%2B2y%2B1%29+=+225
9x%5E2+%2B+54+x+%2B+81+%2B+25y%5E2%2B50y%2B25+=+225
9x%5E2+%2B+54+x+%2B+81+%2B+25y%5E2%2B50y%2B25+=+225
9+x%5E2+%2B+54x+%2B+25y%5E2+%2B+50y+%2B+106=+225
9+x%5E2+%2B+54x+%2B+25y%5E2+%2B+50y+%2B+106-225=0
9x%5E2+%2B+25y%5E2+%2B+54x+%2B+50y+-+119+=+0=> option d.

d. (9x)2 + (25y)2 + 54x + 50y - 119 = 0 -> should be 9x^2 + 25y^2 + 54x + 50y - 119 = 0