Tough question?

The equation for an ellipse centered at the origin is

(x^2 / a^2) + (y^2 / b^2) = 1, where 2a and 2b are the two axes. Assuming your ellipse is wider than it is tall, then a > b, and a = 1.355b. So the equation for your ellipse is always:

(x^2 / 1.836025b^2) + (y^2 / b^2) = 1

For some parameter b.

The y value for the y-intercepts will be when x is 0, so this is when y = ±b.

If (x1, y1) is a point on the ellipse, then you can solve this for b:

(x1^2 / 1.836025b^2) + (y1^2 / b^2) = 1

(x1^2 / 1.836025) + (y1^2) = b^2

b = ±√[ (x1^2 / 1.836025) + (y1^2)]

So for the point (x1,y1) = (3,2), just plug in those points into the above expression for b, and you'll have your y intercept values.

Again, I'm making the assumption that your ellipses are wider than they are tall. But if your major axis was actually along the y axis instead of the x axis, then you could take a = b/1.355 and follow the same line of reasoning to get a different equation.

let's put the major axis on the x-axis. with center on origin, the equation is

x²/a² + y²/b² = 1, and we have a = 1.355b, and (3,2) = (x,y), so

x²/(1.355b)² + y²/b² = 1

9/(1.836025b²) + 4/b² = 1

4.90189 + 4 = b²

b² = 8.90189

b = 2.9836

and that's the y-intercept.

If you need the long axis on the y-axis, let b = 1.355a and solve

3²/a² + 2²/(1.355a)² = 1

9 + 4/1.836025 = a²

a² = 11.1786

a = 3.3434

b = (1.355)(3.3434)

b = 4.53

and THAT will be the y-intercept.

Ellipses are written in the form

(x/a)² + (y/b)² = 1

The y intercept where x is 0. That is, the y intercept is b

You have said that

(3/a)² + (2/b)² = 1

Also, you have pointed out that you know that b/a = some constant k (either 1.355 or its reciprocal, depending on orientation).

Therefore, multiply through by b² to get:

(3b/a)² + 2² = b²

(3k)² + 2² = b²

Thus, the y intercept is

b = √(9k²+4)

In your case, this evaluates to either

4.53 (k=1.355) or 2.98 (k=1/1.355)

First of all, you need to know which axis is larger than the other.

(x/a)^2 + (y/b)^2 = 1

is the general eqn for an ellipse about the origin.

a = 1.355b

OR b = 1.355a depending on which axis is larger.

Then find the values of a and b by substituting (3,2) in the equation.

let the minor axis be b, then the major axis a is 1.355 b

the equation of the ellipse is:

x^2/1.84b^2 + y^2/b^2=1 or x^2/1.84+y^2=b^2.

if it passes through tht point (3,2), then

b^2 =9/1.84+4 =8.89, so b=2.98

to find the y intercept:

put x=0y^2 =8.89or y=2.98