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 nine is equal to one, 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 one, 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 five. 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 nine. If I take the square root on both sides of this equation, we'll get that b is the square root of nine, which is three. So that means that the a axis is five and the b axis is three, and that is the answer to this problem. So I hope you found this helpful, let's move on.

16. Parametric Equations & Polar Coordinates

Conic Sections

Calculus

16. Parametric Equations & Polar Coordinates

Conic Sections

Calculus 때문에 힘들어하고 계신가요?

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객관식

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

$a=25,b=9$

$a=9,b=25$

$a=5,b=3$

$a=3,b=5$

검증된 단계별 안내

Step 1: Recognize the standard form of the hyperbola equation: $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $. Here, $ a^2 = 25 $ and $ b^2 = 9 $.

Step 2: To find the values of $ a $ and $ b $, take the square root of $ a^2 $ and $ b^2 $. This gives $ a = \sqrt{25} $ and $ b = \sqrt{9} $.

Step 3: The length of the $ a $-axis (transverse axis) is calculated as $ 2a $, since the hyperbola is centered at the origin and symmetric about the x-axis.

Step 4: The length of the $ b $-axis (conjugate axis) is calculated as $ 2b $, since the hyperbola is symmetric about the y-axis.

Step 5: Substitute the values of $ a $ and $ b $ into the formulas for the axis lengths to determine the final lengths of the $ a $-axis and $ b $-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.