Question

How do you write an equation of an ellipse in standard form with foci (8, 0) and (-8, 0) if the minor axis has y-intercepts of 2 and -2?

Answer 1

Use x- and y-intercepts to deduce standard form ellipse coefficients. Deduce x-intercepts from distance from the foci to known points.

Explanation:

We can see from the symmetry of the given points that this ellipse is centred at `(x,y)=(0,0)`. So the standard form of the ellipse equation is `x^2/a^2+y^2/b^2=1`.

When `x=0`, we know that `y=+-2`, so `y^2=4`, which gives us `b^2=4`.

A definition of an ellipse is the set of points the sum of whose distances from the two foci is a constant. We can from the given information deduce that distance sum - the distance from both foci to either y-intercept is `sqrt(8^2+2^2)=sqrt(68)`, so the sum is twice that, `2sqrt(68)`.

Let the positive x-intercept be at `x=x_i`. The sum of the distances of this from the foci is `x_i+8+x_i-8=2x_i`, which is equal to `2sqrt(68)`, so `x_i=sqrt(68)`. At these points `y=0`, so we deduce that `68/a^2=1`, i.e. `a^2=68`.

Thus
`x^2/68+y^2/4=1`.