Given the hyperbola x2−y24=1x^2-\frac{y^2}{4}=1x2−4y2=1, find the length of the aaa -axis and bbb -axis.

a = 1 a=1 a = 1 , b = 2 b=2 b = 2

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

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

a = 1 a=1 a = 1 , b = 2 b=2 b = 2

a = 1 a=1 a = 1 , b = 4 b=4 b = 4

a = 4 a=4 a = 4 , b = 1 b=1 b = 1

a = 2 a=2 a = 2 , b = 1 b=1 b = 1

19. Conic Sections

Hyperbolas at the Origin

Precalculus

19. 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. The given equation is $ x^2 - \frac{y^2}{4} = 1 $, which can be rewritten as $ \frac{x^2}{1} - \frac{y^2}{4} = 1 $.

Recognize that this equation is in the form $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $, which represents a hyperbola centered at the origin with a horizontal transverse axis.

From the equation $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $, identify $ a^2 = 1 $ and $ b^2 = 4 $.

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

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

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