Given the hyperbola x 2 25 − y 2 9 = 1 \frac{x^2}{25}-\frac{y^2}{9}=1 , find the length of the a a -axis and the b b -axis.

Hey everyone. Let's see if we can solve this problem. So it says here that given the hyperbola, x squared over 25, minus y squared over 9 is equal to 1, find the length of the a and b axis. So to solve this problem, we need to recognize the standard form for a hyperbola. And what we know is that we have x squared over a squared, minus minus y squared over b squared is equal to 1. Assuming that we have a horizontally oriented hyperbola with the a underneath the x. Now what I can see here is that we do have this type of situation, and recall the a squared is always what comes first. So what I can do is I can say that a squared is equal to what we have first in the denominator which is 25, then you take the square root on both sides of this equation, you get that a is the square root of 25 which is 5. Now what I can do next is take our b squared, and set that equal to the second term we see in the denominator, which is 9. If I take the square root on both sides of this equation, we'll get that b is the square root of 9 which is 3. So that means that the a axis is 5, and the b axis is 3, and that is the answer to this problem. So I hope you found this helpful, let's move on.

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 $\frac{x^{2}}{25} - \frac{y^{2}}{9} = 1$ , find the length of the $a$ -axis and the $b$ -axis.

$a = 25, b = 9$

$a = 9, b = 25$

$a = 5, b = 3$

$a = 3, b = 5$

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, $ a^2 = 25 $ and $ b^2 = 9 $.

To find the length of the transverse axis (a-axis), calculate $ 2a $. First, find $ a $ by taking the square root of $ a^2 $: $ a = \sqrt{25} = 5 $. Therefore, the length of the transverse axis is $ 2a = 2 \times 5 = 10 $.

To find the length of the conjugate axis (b-axis), calculate $ 2b $. First, find $ b $ by taking the square root of $ b^2 $: $ b = \sqrt{9} = 3 $. Therefore, the length of the conjugate axis is $ 2b = 2 \times 3 = 6 $.

Verify the orientation of the hyperbola. Since the $ x^2 $ term is positive, the hyperbola opens horizontally, confirming that $ a $ corresponds to the transverse axis and $ b $ to the conjugate axis.

Summarize the results: The length of the transverse axis (a-axis) is 10, and the length of the conjugate axis (b-axis) is 6.

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Struggling with College Algebra?

Given the hyperbola x225−y29=1\frac{x^2}{25}-\frac{y^2}{9}=1, find the length of the aa-axis and the bb-axis.

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Given the hyperbola $\frac{x^{2}}{25} - \frac{y^{2}}{9} = 1$ , find the length of the $a$ -axis and the $b$ -axis.