Find k for Ellipse with Major Axis Length 6 | Exam Study Help

[petuniac]

ellipse - urgent help needed

Hi there.. i have an exam tomorrow morning and can't figure this out.

ellipse defined by 3x^2 + 2y^2 = k
length of major axis is 6
find k?

ok.. i know that the major axis = 2a, so in this case a = 3
the general form of an ellipse is

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

? where do i go now ?

 

[Ateowa]

So the equation for the ellipse is 3x² + 2y²=k. The length of the major axis let's you fill in the following into the ellipse equation: x²/9 + y²/b^2=1. The 9 allows you to finish the problem. Do you remember how you bring an ellipse from the general form to the standard form for an ellipse?

Let's pretend that we're trying to take the original equation to standard form for an ellipse. You want the right side of the equation to equal 1, so you divide by k. That means that the standard form, in this case, would be 3x²/k + 2y²/k=1. But you always have to simplify. The coefficient of x must be gone in the final equation, so that means that the a² must have been simplified. With that information, you should be able to take the jump to find k (That is, if I worded it right =) ).

 

[Architeuthis Dux]

I would first start by clarifying the question. Is the given ellipse in standard form or general form? This will affect the approach to finding k.

If the given ellipse is in standard form, then we can directly compare it to the general form and determine that a = 3 and b = √(k/2). Since the length of the major axis is 6, we know that 2a = 6, which means a = 3. Therefore, we can substitute a = 3 into the general form to get:

x^2/3^2 + y^2/b^2 = 1

Simplifying, we get:

x^2/9 + y^2/b^2 = 1

Comparing this to the given equation, 3x^2 + 2y^2 = k, we can see that b^2 = 2k/3. Since we know that the length of the major axis is 6, we can set 2a = 6 and solve for b to get:

2a = 6
2(3) = 6
b = √(2k/3)
2(3) = √(2k/3)
36 = 2k/3
k = 54

Therefore, the value of k for this ellipse is 54.

However, if the given ellipse is in general form, then we can use the formula for the length of the major axis, 2a = √(a^2 + b^2), to solve for k. Since we know that the length of the major axis is 6 and a = 3, we can set up the following equation:

2a = √(a^2 + b^2)
2(3) = √(3^2 + b^2)
6 = √(9 + b^2)
36 = 9 + b^2
b^2 = 27

Therefore, the value of k for this ellipse is 3x^2 + 2y^2 = 27.

In summary, the approach to finding k for this ellipse will depend on whether it is in standard form or general form. Once this is clarified, we can use the given information to solve for k using the appropriate method. I hope this helps and good luck on your exam tomorrow!