Question 656908
Write the Equation of the ellipse that meet each set of conditions.. 
40. The foci are at (3,5) and (1,5) and the ellipse has eccentricity of 0.25 
41. The ellipse has a vertical major axis of 20 units, its center is at (3,0) and e= 7/10 (eccentricity)
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40. The foci are at (3,5) and (1,5) and the ellipse has eccentricity of 0.25.
This is an ellipse with horizontal major axis.
Its standard form of equation: {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}, a>b, (h,k)=(x,y) coordinates of center.
For given ellipse:
center: (2,5)
length of horizontal major axis=2 (1 to 3)=2a
a=1
a^2=1
..
eccentricity=c/a=.25=1/4
c=a/4=1/4
c^2=1/16
..
c^2=a^2-b^2
b^2=a^2-c^2=1-1/16=15/16
..
Equation of ellipse:
{{{(x-2)^2/1+(y-5)^2/(15/16)=1}}}
{{{(x-2)^2/1+16(y-5)^2/15=1}}}
..
41. The ellipse has a vertical major axis of 20 units, its center is at (3,0) and e= 7/10 (eccentricity)
This is an ellipse with vertical major axis as stated
Its standard form of equation: {{{(x-h)^2/b^2+(y-k)^2/a^2=1}}}, a>b, (h,k)=(x,y) coordinates of center.
For given ellipse:
center: (3,0)
length of vertical major axis=20=2a
a=10
a^2=100
..
eccentricity=c/a=7/10
c=7a/10=7
c^2=49
..
c^2=a^2-b^2
b^2=a^2-c^2=100-49=51
..
Equation of ellipse:
{{{(x-3)^2/51+y^2/100=1}}}