Equation of the Ellipse?

Question

Find the equation of the ellipse with x intercepts at (6,0) an y intercepts at (0,4). Find the foci.

Not sure where to start. Please help

10 points

Answer 1

the answer above is exactly correct

in case you wanted a second opinion

Answer 2

If the standard ellipse has x-intercept a and y-intercept b, its equation is

x²/a² + y²/b² = 1

Foci are (±ae, 0) if a > b and (0, ±be) if b > a.

where e is the eccentricity given by

e² = (a² - b²)/a² if a > b and

e² = (b² - a²)/b² if b > a

In the above example, equation of the ellipse is

x²/36 + y²/16 = 1

and as a = 6 > b = 4,

e² = (36 - 16)/36 = 20/36 = 5/9 > e = √5/3

=> foci are (±2√5, 0).

Answer 3

I'm guessing that there are an infinite number of ellipses through any two points.

I guess that your ellipse must be centred on the origin & have its axis parallel to the x & y axes.

So the intercepts (well two of them) tell you that when x = 0, y = 4 & when y = 0, x = 6.

Since the ellipse seems to be wider than it is tall, the foci must be points on the x axis. & symmetry says they must be symmetrically placed about the origin.

The distance from one focus to a point on the outside plus the distance from that point to the other focus is a constant, so geometry should give the foci to you.  Maybe take a couple of points on the curve that give you two equations & solve.  Take some easy points, like the intercepts?

Any help?

Answer 4

The equation of the ellipse is x^2/a^2 + y^2/b^2 = 1

Given intercept x=6 y=0 it means a=6

Given intercept x=0 y=4 it means b=4

the equation is x^2/36 + y^2/16 =1

Foci should be figured out from c^2=a^2-b^2= sqrt20=2*sqrt5

Foci are (+_ 2*sqrt5; 0)

Answer 5

you have only given one intercept for the x and y axis, so I am assuming that the two x intercepts are at (6,0) and (-6,0), and the y intercepts are at (0,4) and (0,-4)

this means that the semi-major axis (half the long way across the ellipse) is 6, and the semi minor axis (half the short way across the ellipse) is 4

the coordinates of the center of the ellipse are (0,0)

so...

the equation for an ellipse is:

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

where...

(h,k) are the coordinates of the center of the ellipse

a= semi major axis

b= semi minor axis

here:

(h,k)=(0,0)

a=6

b=4

so the equation of this ellipse is

x^2/36 + y^2/16=1