Given the hyperbola x225−y29=1\frac{x^2}{25}-\frac{y^2}{9}=125x2−9y2=1, find the length of the aaa-axis and the bbb -axis.

a = 5 , b = 3 a=5,b=3 a = 5 , b = 3

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

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

a = 5 , b = 3 a=5,b=3 a = 5 , b = 3

a = 25 a=25 a = 25 , b = 9 b=9 b = 9

a = 9 , b = 25 a=9,b=25 a = 9 , b = 25

a = 3 , b = 5 a=3,b=5 a = 3 , b = 5

19. Conic Sections

Hyperbolas at the Origin

Precalculus

19. 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 $.

Calculate the value of $ a $ by taking the square root of $ a^2 $: $ a = \sqrt{25} = 5 $.

Calculate the value of $ b $ by taking the square root of $ b^2 $: $ b = \sqrt{9} = 3 $.

The length of the $ a $-axis (transverse axis) is $ 2a $, which is $ 2 \times 5 = 10 $.

The length of the $ b $-axis (conjugate axis) is $ 2b $, which is $ 2 \times 3 = 6 $.

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