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

1 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 .