Given the hyperbola y2100−x2139=1\frac{y^2}{100}-\frac{x^2}{139}=1100y2−139x2=1, find the length of the aaa-axis and the bbb-axis.

a = 10 , b = 139 a=10,b=\sqrt{139} a = 10 , b = 139 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Given the hyperbola y 2 100 − x 2 139 = 1 \frac{y^2}{100}-\frac{x^2}{139}=1 100 y 2 − 139 x 2 = 1 , find the length of the a a a -axis and the b 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 one thirty nine is equal to one, and 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 one. Now what I can do here is recognize that a squared is equal to 100. So we'll 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 one thirty nine. 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 one thirty nine. 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 two 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}=1 100 y 2 − 139 x 2 = 1 , find the length of the a a a -axis and the b b b -axis.

a = 10 , b = 139 a=10,b=\sqrt{139} a = 10 , b = 139 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

a = 100 , b = 139 a=100,b=139 a = 100 , b = 139

a = 139 , b = 100 a=139,b=100 a = 139 , b = 100

a = 139 , b = 10 a=\sqrt{139},b=10 a = 139 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , b = 10

16. Parametric Equations & Polar Coordinates

Conic Sections

Calculus

16. Parametric Equations & Polar Coordinates

Conic Sections

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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

Step 1: Recognize the standard form of the hyperbola equation. The given equation is $ \frac{y^2}{100} - \frac{x^2}{139} = 1 $, which represents a vertical hyperbola because the $ y^2 $ term is positive and comes first.

Step 2: Identify the values of $ a^2 $ and $ b^2 $ from the denominators of the equation. Here, $ a^2 = 100 $ and $ b^2 = 139 $.

Step 3: Calculate $ a $ and $ b $ by taking the square root of $ a^2 $ and $ b^2 $. This gives $ a = \sqrt{100} = 10 $ and $ b = \sqrt{139} $.

Step 4: Understand the geometric significance of $ a $ and $ b $. For a vertical hyperbola, $ a $ represents the distance from the center to the vertices along the $ y $-axis, and $ b $ represents the distance from the center to the co-vertices along the $ x $-axis.

Step 5: The lengths of the $ a $-axis and $ b $-axis are determined as follows: The $ a $-axis length is $ 2a $, which equals $ 2 \times 10 = 20 $, and the $ b $-axis length is $ 2b $, which equals $ 2 \times \sqrt{139} $.

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