Find an ellipse centered through the origin that runs through 3 points

[mahrap]

Find the ellipse centered at the origin that runs through
the points (1,2), (2,2), and (3, I). Write your equation
in the form $$ ax^2 + bxy + cy^2 = 1 $$

I understand the $$ ax^2 $$ and $$ cy^2 $$ in the equation because the equation of an ellipse centered at origin is $$ (x/a)^2 + (y/b)^2 = 1 $$ so we let $$ a = (1/a)^2 $$ and $$ b = (1/b)^2 $$. but where did the $$ bxy $$ come from?

 

[Dick]

Find the ellipse centered at the origin that runs through
the points (1,2), (2,2), and (3, I). Write your equation
in the form $$ ax^2 + bxy + cy^2 = 1 $$

I understand the $$ ax^2 $$ and $$ cy^2 $$ in the equation because the equation of an ellipse centered at origin is $$ (x/a)^2 + (y/b)^2 = 1 $$ so we let $$ a = (1/a)^2 $$ and $$ b = (1/b)^2 $$. but where did the $$ bxy $$ come from?

$(x/a)^2 + (y/b)^2 = 1$ is only the equation of an ellipse centered at the origin whose axes are parallel to x and y axes. $ax^2 + bxy + cy^2 = 1$ is more general. It may be at an angle. Just put the given values for x and y in and get three equations to solve for the three unknowns a, b and c.