How do you write an equation of an ellipse in standard form given foci (+/-4,0) and co-vertices at (0,+/-2)?

1 Answer

Answer:

#x^2/20+y^2/4=1#

Explanation:

From the given data:

Foci(4, 0) and (-4, 0)
Co-vertices (0, 2) and (0, -2) are the endpoints of the minor axis.

The Center is at the Origin #(0, 0)# by inspection.

The standard form equation for Horizontal major axis Ellipse

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

#b=2# the length of 1/2 of the minor axis called semi-minor axis.
#c=4# the distance from center to a focus.

#a# is the length of 1/2 of the major axis also called semi-major axis.

Compute for #a#

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

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

#a=sqrt(4^2+2^2)#

#a=sqrt(16+4)#

#a=sqrt(20)=2sqrt(5)#

Use now the equation

#(x-h)^2/a^2+(y-k)^2/b^2=1# with Center #(h, k)=(0, 0)#

#(x-0)^2/(2sqrt(5))^2+(y-0)^2/2^2=1#

which simplifies to

#x^2/20+y^2/4=1#

Kindly check the graph...

graph{(x^2/20+y^2/4-1)=0[-10,10,-5,5]}

Have a nice day !!! from the Philippines .