How do you write an equation of an ellipse for the given Foci (0,±8) Co-Vertices (±8,0)?

1 Answer

Answer:

#x^2/64+y^2/128=1#

Explanation:

From the given:
Foci: #F_1(0, 8), F_2(0, -8)#
Co-vertices at #(8, 0) # and (-8, 0)#

by inspection, the center of the ellipse is at Origin #(0, 0)#
that means

Center #(h, k)=(0, 0)#

Also, by inspection, #c=8# the distance from the center to a focus.
Also, #b=8# the distance from the center to one end point of the minor axis also called semi-minor axis length.

Compute for semi-major axis length #a#:

#a^2=b^2+c^2#

#a=sqrt(b^2+c^2)#

#a=sqrt(8^2+8^2)=sqrt(64+64#

#a=sqrt(2*64)#

#a=8sqrt(2)#

Use now the standard Form of the equation of ellipse for Vertical Major Axis.

#(x-h)^2/b^2+(y-k)^2/a^2=1#

Let us put on the values of #h=0, k=0, a=8sqrt(2), b=8#

#(x-0)^2/8^2+(y-0)^2/(8sqrt(2))^2=1#

#x^2/64+y^2/128=1#

Kindly check the graph ....
graph{(x^2/64+y^2/128-1)=0[-25,25,-15,15]}

Have a nice day !!! from the Philippines...