What is the sum of l1 and l2 for every horizontal ellipse?

An ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. An ellipse usually looks like a squashed circle. From the figure below F is a focus, G is a focus. Together they are called foci. So the sum of L1 and L2 is always the same value, that is, if we go from point F to any point on the ellipse and then go on to point G, we always travel the same distance. This happens for every horizontal ellipse as indicated in the Figure below. In mathematical language:

hello from mrbilldoesmath!

answer:

(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)

discussion:

the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,

a^3 + b^3 =   (a + b) (a^2 - a b + b^2).

in our case

343x^3+216y^3   =   (7x)^3 + (6y)^3

so take a = 7x and b = 6y in the above equation to get the answer.

you,

mrb