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?

1 Answer
Jun 1, 2018

Answer:

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#.