Find an equation of the form ax2 + by2 = c for the ellipse that passes through the given points. (3, 4) and (6, 2)

Question

Find an equation of the form ax² + by² = c

 for the ellipse that passes through the given points.

(3, 4) and (6, 2)

Answer 1

an ellipse through (3,4) and (6,2)

would satisfy

9a + 16b = c

36a + 4b = c

that's 2 equations 3 unknowns with an infinite number of possible ellipses

36a + 64b = 4c

36a +4b = c

60b = 3c

20b = c, let c=20, then b=1, a =(c-4b)/36 = 16/36 = 4/9

ellipse is 4x^2/9 +y^2 = 20

or 4x^2 +9y^2 = 180 among an infinite number of other ellipses