Math / Geometry

QuestionWhat is the length of the major axis of the conic section shown below? $\frac{(x+2)^{2}}{64}+\frac{(y-1)^{2}}{81}=1$

Studdy Solution

STEP 1

What is this asking? How long is the long diameter of this oval? Watch out! Don't mix up the a and b values!
Remember which one goes with which axis!
Also, remember that the equation gives us $a^2$ and $b^2$ , not a and b directly!

STEP 2

1. Identify the conic section.
2. Find the major and minor axes.
3. Calculate the length of the major axis.

STEP 3

The equation is in the form $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .
This tells us we're dealing with an ellipse!
It's centered at $(-2, 1)$ , but that doesn't matter for this problem.

STEP 4

We see that $a^2 = 64$ and $b^2 = 81$ .
Taking the square root of both sides gives us $a = 8$ and $b = 9$ .
Remember, we only care about the positive root since lengths can't be negative!

STEP 5

Since $b > a$ , the major axis is vertical and has length $2b$ , and the minor axis is horizontal and has length $2a$ .
We know this because the bigger number is under the $y$ term, and $y$ controls vertical movement.

STEP 6

The major axis has length $2b$ .
We found that $b = 9$ , so the length of the major axis is $2 \cdot 9 = 18$ .
Boom!

STEP 7

The length of the major axis of the ellipse is 18.

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QuestionWhat is the length of the major axis of the conic section shown below? $\frac{(x+2)^{2}}{64}+\frac{(y-1)^{2}}{81}=1$

Studdy Solution

STEP 1

What is this asking? How long is the long diameter of this oval? Watch out! Don't mix up the a and b values!
Remember which one goes with which axis!
Also, remember that the equation gives us $a^2$ and $b^2$ , not a and b directly!

STEP 2

1. Identify the conic section.
2. Find the major and minor axes.
3. Calculate the length of the major axis.

STEP 3

The equation is in the form $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .
This tells us we're dealing with an ellipse!
It's centered at $(-2, 1)$ , but that doesn't matter for this problem.

STEP 4

We see that $a^2 = 64$ and $b^2 = 81$ .
Taking the square root of both sides gives us $a = 8$ and $b = 9$ .
Remember, we only care about the positive root since lengths can't be negative!

STEP 5

Since $b > a$ , the major axis is vertical and has length $2b$ , and the minor axis is horizontal and has length $2a$ .
We know this because the bigger number is under the $y$ term, and $y$ controls vertical movement.

STEP 6

The major axis has length $2b$ .
We found that $b = 9$ , so the length of the major axis is $2 \cdot 9 = 18$ .
Boom!

STEP 7

The length of the major axis of the ellipse is 18.

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QuestionWhat is the length of the major axis of the conic section shown below? (x+2)264+(y−1)281=1\frac{(x+2)^{2}}{64}+\frac{(y-1)^{2}}{81}=164(x+2)2​+81(y−1)2​=1

Studdy Solution

STEP 1

What is this asking? How long is the long diameter of this oval? Watch out! Don't mix up the *a* and *b* values!Remember which one goes with which axis!Also, remember that the equation gives us a2a^2a2 and b2b^2b2, not *a* and *b* directly!

STEP 2

1. Identify the conic section.2. Find the major and minor axes.3. Calculate the length of the major axis.

STEP 3

The equation is in the form (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1a2(x−h)2​+b2(y−k)2​=1.This tells us we're dealing with an **ellipse**!It's centered at (−2,1)(-2, 1)(−2,1), but that doesn't matter for this problem.

STEP 4

We see that a2=64a^2 = 64a2=64 and b2=81b^2 = 81b2=81.Taking the square root of both sides gives us a=8a = 8a=8 and b=9b = 9b=9.Remember, we only care about the *positive* root since lengths can't be negative!

STEP 5

Since b>ab > ab>a, the **major axis** is vertical and has length 2b2b2b, and the **minor axis** is horizontal and has length 2a2a2a.We know this because the bigger number is under the yyy term, and yyy controls vertical movement.

STEP 6

The major axis has length 2b2b2b.We found that b=9b = 9b=9, so the length of the major axis is 2⋅9=182 \cdot 9 = 182⋅9=18.Boom!

STEP 7

The length of the major axis of the ellipse is **18**.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionWhat is the length of the major axis of the conic section shown below? $\frac{(x+2)^{2}}{64}+\frac{(y-1)^{2}}{81}=1$

What is this asking? How long is the long diameter of this oval? Watch out! Don't mix up the a and b values!
Remember which one goes with which axis!
Also, remember that the equation gives us $a^2$ and $b^2$ , not a and b directly!

1. Identify the conic section.
2. Find the major and minor axes.
3. Calculate the length of the major axis.

The equation is in the form $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .
This tells us we're dealing with an ellipse!
It's centered at $(-2, 1)$ , but that doesn't matter for this problem.

We see that $a^2 = 64$ and $b^2 = 81$ .
Taking the square root of both sides gives us $a = 8$ and $b = 9$ .
Remember, we only care about the positive root since lengths can't be negative!

Since $b > a$ , the major axis is vertical and has length $2b$ , and the minor axis is horizontal and has length $2a$ .
We know this because the bigger number is under the $y$ term, and $y$ controls vertical movement.

The major axis has length $2b$ .
We found that $b = 9$ , so the length of the major axis is $2 \cdot 9 = 18$ .
Boom!

The length of the major axis of the ellipse is 18.