Ellipse through (1,6) and (2,3)

An ellipse with centre at the origin and axes along the coordinate axes has the form

x²/a² + y²/b² = 1

Let A = 1/a² and B = 1/b². Then Ax² + By² = 1.

Use the two points

(1, 6): A + 36B = 1 ... (i) (2, 3): 4A + 9B = 1 ... (ii)

From (i), A = 1 − 36B. Substituting in (ii):

4 − 144B + 9B = 1 135B = 3, so B = 1/45

A = 1 − 36/45 = 9/45 = 1/5

So a² = 5 and b² = 45.

Required ellipse: x²/5 + y²/45 = 1, i.e. 9x² + y² = 45.

Check: (1, 6): 9 + 36 = 45 ✓. (2, 3): 36 + 9 = 45 ✓.

Here b² > a², so the major axis lies along the y-axis (semi-major 3√5, semi-minor √5).