Given the hyperbola x 2 − y 2 4 = 1 x^2-\frac{y^2}{4}=1 , find the length of the a a -axis and b b -axis.

Hey everyone. Let's see if we can solve this problem. So here we have the hyperbola x squared minus y squared over 4 is equal to 1, and we're asked to find the length of the a and b axis. So what I can do by looking at this problem is recognize that we have this form of a hyperbola. It's x squared over a squared minus y squared over b squared is equal to 1. And by looking at the standard form of the hyperbola, I can see where things are going to line up. Now this a squared is going to correspond with what's in the denominator of this x squared. But notice that this x squared doesn't have a denominator, so we can just imagine that there's a one. So that means that a squared is equal to 1. And what I can do is take the square root on both sides to get that a is 1. So that's a. And for b squared, b squared is associated with this 4. So you have the b squared is equal to 4, and then if I take the square root on both sides, b is equal to the square root of 4 which is 2. So this is our a and b axis, and notice in this case that even though the b value was larger, it still doesn't become the a value. The a value is simply whatever comes first, and since this one came first, that's what we set a squared equal to, and that's how we got a. That's the answer. These are the a and b axes. Hope you found this helpful.

8. Conic Sections

Hyperbolas at the Origin

College Algebra

8. Conic Sections

Hyperbolas at the Origin

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Multiple Choice

Given the hyperbola $x^{2} - \frac{y^{2}}{4} = 1$ , find the length of the $a$ -axis and $b$ -axis.

$a = 1, b = 4$

$a = 4, b = 1$

$a = 1, b = 2$

$a = 2, b = 1$

Verified step by step guidance

Identify the standard form of the hyperbola equation: $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $. In this case, the given equation is $ x^2 - \frac{y^2}{4} = 1 $.

Compare the given equation with the standard form to determine the values of $ a^2 $ and $ b^2 $. Here, $ a^2 = 1 $ and $ b^2 = 4 $.

Calculate the value of $ a $ by taking the square root of $ a^2 $. Since $ a^2 = 1 $, $ a = \sqrt{1} = 1 $.

Calculate the value of $ b $ by taking the square root of $ b^2 $. Since $ b^2 = 4 $, $ b = \sqrt{4} = 2 $.

The length of the transverse axis (a-axis) is $ 2a = 2 \times 1 = 2 $ and the length of the conjugate axis (b-axis) is $ 2b = 2 \times 2 = 4 $.

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Given the hyperbola $x^{2} - \frac{y^{2}}{4} = 1$ , find the length of the $a$ -axis and $b$ -axis.