Tough question?

Question

I have an ellipse who's proportions are always the same: the major axis is always 1.355 times the minor axis. It is centered on and symmetrical about the origin (0,0).

Given that the ellipse passes through the point (3,2), what is the value of the y-intercept?

Actually, I don't need the answer, I just need to know how to solve for the y-intercept. This is a work problem that has been plaguing me for days.

Answer 1

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.

Answer 2

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.

Answer 3

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)

Answer 4

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.

Answer 5

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