Given the hyperbola y2100−x2139=1\frac{y^2}{100}-\frac{x^2}{139}=1, find the length of the aa-axis and the bb-axis.

a = 10 , b = 139 a=10,b=\sqrt{139}

Given the hyperbola y 2 100 − x 2 139 = 1 \frac{y^2}{100}-\frac{x^2}{139}=1 , find the length of the a a -axis and the b b -axis.

Welcome back everyone. Let's give this problem a try. So here we are given the hyperbola y squared over a 100 minus x squared over 139 is equal to 1. We're asked to find the length of the a and b axis. Now to solve this problem, notice that it's the y squared term that comes first. So that means we're dealing with a vertically oriented hyperbola, and the equation for a vertical hyperbola is y squared over a squared minus x squared over b squared is equal to 1. Now what I can do here is recognize that a squared is equal to 100. So I have a squared equals a 100, and if I take the square root on both sides of this equation, we'll get that a is equal to the square root of a 100, which is 10. Now what I can also recognize is that b squared is equal to 139. And to solve for b, I can just take the square root on both sides of this equation, canceling the square, giving us that b is simply the square root of 139. This does not simplify down any farther. So this would be our b value, and that would be a. And that is how you can find the 2 axes, so that's the answer to the problem. Hope you found this video helpful, thanks for watching.

Given the hyperbola y 2 100 − x 2 139 = 1 \frac{y^2}{100}-\frac{x^2}{139}=1 , find the length of the a a -axis and the b b -axis.

a = 10 , b = 139 a=10,b=\sqrt{139}

a = 139 , b = 10 a=\sqrt{139},b=10

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 $\frac{y^{2}}{100} - \frac{x^{2}}{139} = 1$ , find the length of the $a$ -axis and the $b$ -axis.

$a = 100, b = 139$

$a = 139, b = 100$

$a = \sqrt{139}, b = 10$

$a = 10, b = \sqrt{139}$

Verified step by step guidance

Identify the standard form of the hyperbola equation: $ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 $. This indicates a vertical hyperbola because the $ y^2 $ term is positive.

Compare the given equation $ \frac{y^2}{100} - \frac{x^2}{139} = 1 $ with the standard form to determine $ a^2 = 100 $ and $ b^2 = 139 $.

Calculate $ a $ by taking the square root of $ a^2 $: $ a = \sqrt{100} = 10 $.

Calculate $ b $ by taking the square root of $ b^2 $: $ b = \sqrt{139} $.

The length of the transverse axis (a-axis) is $ 2a = 2 \times 10 = 20 $ and the length of the conjugate axis (b-axis) is $ 2b = 2 \times \sqrt{139} $.

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