Answer:
Explanation:
The standard form of the ellipse equation is (x-h)²/a² +(y-k)²/b² =1
This equation is for a horizontal ellipse. (a) is always bigger than (b) in ellipses. Because (a²) is under the x, that's why this ellipse is horizontal.
Now, let's answer the questions:
1. The center coordinates are (h,k) which are (4,-2) by comparing the two equations.
2. length of the major axis = 2a , let's find (a)
if you compare the two equations, a²=36 , that means a=6
Then 2a=12
3. length of the minor axis = 2b , let's find (b)
if you compare the two equations, b²=16 , that means b=4
Then 2b=8
4. the sum of the local radii = length of the major axis = 12
5. to find the foci, we have to find the value of (c), we have to use the equation a²=b²+c²
plugin a² and b², 36=16+c² , solve it, c=√20 =2√5
Now, to find the coordinates of the foci, we have to add and subtract the (c) value from the x-coordinate of the center while the y-coordinate stays the same (because this ellipse is horizontal).
the coordinates of the foci are (4+2√5 , -2) and (4-2√5 , -2)
6. the vertices are the end points of the major and the minor axes. To find the vertices on the major axis, add the value of (a) to the x-coordinate of the center while the y-coordinate stays the same.
(4+6,-2) and (4-6,-2) , this leads to (10,-2) and (-2,-2)
To find the vertices on the minor axis, add the value of (b) to the y-coordinate of the center while the x-coordinate stays the same.
(4,-2+4) and (4,-2-4) , this leads to (4,2) and (4,-6)
7. eccentricity (e)=c/a =(2√5)/6 = (√5)/3