Math / Geometry

QuestionGiven the general form of an ellipse, $4 x^{2}+y^{2}-24 x+8 y+16=0$ , determine the correct value for each characteristic.
Orientation: [ Select ]
Center $(h, k)$ : $h=$ [Select ] $k=$ [ Select ]
Length of the major axis: [ Select ]

Studdy Solution

STEP 1

What is this asking? We're given an equation for an ellipse and need to find its orientation, center, and the length of its major axis! Watch out! Don't mix up the major and minor axes – the major axis is the longer one!
Also, remember that completing the square involves adding the same value to both sides of the equation.

STEP 2

1. Rewrite the equation
2. Complete the square
3. Analyze the ellipse

STEP 3

Let's group the $x$ terms and $y$ terms together to make completing the square easier!
We'll rewrite the given equation, $4x^2 + y^2 - 24x + 8y + 16 = 0$ , as $4(x^2 - 6x) + (y^2 + 8y) = -16$ .
See how we factored out the coefficient of the squared terms?
This sets us up perfectly for the next step!

STEP 4

Now, let's complete the square for the $x$ terms.
Take half of the coefficient of the $x$ term, which is $\frac{-6}{2} = -3$ , square it to get $(-3)^2 = 9$ , and add it inside the parentheses.
Remember, since we factored out a 4, we're actually adding $4 \cdot 9 = 36$ to the left side, so we must add 36 to the right side as well!
This keeps our equation balanced!

STEP 5

Our equation now looks like $4(x^2 - 6x + 9) + (y^2 + 8y) = -16 + 36$ .
Let's do the same for the $y$ terms!
Half of the coefficient of the $y$ term is $\frac{8}{2} = 4$ , and squaring it gives us $4^2 = 16$ .
Adding 16 to both sides, we get $4(x^2 - 6x + 9) + (y^2 + 8y + 16) = -16 + 36 + 16$ .

STEP 6

Now we can rewrite our equation in a much nicer form: $4(x-3)^2 + (y+4)^2 = 36$ .
Almost there!
To get the standard form of an ellipse equation, we need a 1 on the right side.

STEP 7

Let's divide both sides of the equation by 36 to achieve this.
We get $\frac{4(x-3)^2}{36} + \frac{(y+4)^2}{36} = \frac{36}{36}$ , which simplifies to $\frac{(x-3)^2}{9} + \frac{(y+4)^2}{36} = 1$ .
This is the standard form of an ellipse equation!

STEP 8

From the standard form, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , we can see that the center of our ellipse is $(h, k) = (3, -4)$ .
Since $36 > 9$ , the major axis is vertical (along the $y$ -axis).

STEP 9

The length of the major axis is $2b$ , where $b^2 = 36$ , so $b = 6$ .
Therefore, the length of the major axis is $2 \cdot 6 = 12$ .

STEP 10

Orientation: Vertical Center $(h, k)$ : $h = 3$ , $k = -4$ Length of the major axis: 12

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QuestionGiven the general form of an ellipse, $4 x^{2}+y^{2}-24 x+8 y+16=0$ , determine the correct value for each characteristic.
Orientation: [ Select ]
Center $(h, k)$ : $h=$ [Select ] $k=$ [ Select ]
Length of the major axis: [ Select ]

Studdy Solution

STEP 1

STEP 2

1. Rewrite the equation
2. Complete the square
3. Analyze the ellipse

STEP 3

Let's group the $x$ terms and $y$ terms together to make completing the square easier!
We'll rewrite the given equation, $4x^2 + y^2 - 24x + 8y + 16 = 0$ , as $4(x^2 - 6x) + (y^2 + 8y) = -16$ .
See how we factored out the coefficient of the squared terms?
This sets us up perfectly for the next step!

STEP 4

STEP 5

Our equation now looks like $4(x^2 - 6x + 9) + (y^2 + 8y) = -16 + 36$ .
Let's do the same for the $y$ terms!
Half of the coefficient of the $y$ term is $\frac{8}{2} = 4$ , and squaring it gives us $4^2 = 16$ .
Adding 16 to both sides, we get $4(x^2 - 6x + 9) + (y^2 + 8y + 16) = -16 + 36 + 16$ .

STEP 6

Now we can rewrite our equation in a much nicer form: $4(x-3)^2 + (y+4)^2 = 36$ .
Almost there!
To get the standard form of an ellipse equation, we need a 1 on the right side.

STEP 7

Let's divide both sides of the equation by 36 to achieve this.
We get $\frac{4(x-3)^2}{36} + \frac{(y+4)^2}{36} = \frac{36}{36}$ , which simplifies to $\frac{(x-3)^2}{9} + \frac{(y+4)^2}{36} = 1$ .
This is the standard form of an ellipse equation!

STEP 8

From the standard form, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , we can see that the center of our ellipse is $(h, k) = (3, -4)$ .
Since $36 > 9$ , the major axis is vertical (along the $y$ -axis).

STEP 9

The length of the major axis is $2b$ , where $b^2 = 36$ , so $b = 6$ .
Therefore, the length of the major axis is $2 \cdot 6 = 12$ .

STEP 10

Orientation: Vertical Center $(h, k)$ : $h = 3$ , $k = -4$ Length of the major axis: 12

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QuestionGiven the general form of an ellipse, 4x2+y2−24x+8y+16=04 x^{2}+y^{2}-24 x+8 y+16=04x2+y2−24x+8y+16=0, determine the correct value for each characteristic.Orientation: [ Select ]Center (h,k)(h, k)(h,k) : h=h=h= [Select ] k=k=k= [ Select ]Length of the major axis: [ Select ]

Studdy Solution

STEP 1

STEP 2

1. Rewrite the equation2. Complete the square3. Analyze the ellipse

STEP 3

STEP 4

STEP 5

STEP 6

Now we can rewrite our equation in a much nicer form: 4(x−3)2+(y+4)2=364(x-3)^2 + (y+4)^2 = 364(x−3)2+(y+4)2=36.Almost there!To get the standard form of an ellipse equation, we need a **1** on the right side.

STEP 7

STEP 8

From the standard 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, we can see that the center of our ellipse is (h,k)=(3,−4)(h, k) = (3, -4)(h,k)=(3,−4).Since 36>936 > 936>9, the major axis is vertical (along the yyy-axis).

STEP 9

The length of the major axis is 2b2b2b, where b2=36b^2 = 36b2=36, so b=6b = 6b=6.Therefore, the length of the major axis is 2⋅6=122 \cdot 6 = 122⋅6=12.

STEP 10

Orientation: Vertical Center (h,k)(h, k)(h,k): h=3h = 3h=3, k=−4k = -4k=−4 Length of the major axis: 12

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Studdy solves anything!

Start learning now

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

QuestionGiven the general form of an ellipse, $4 x^{2}+y^{2}-24 x+8 y+16=0$ , determine the correct value for each characteristic.
Orientation: [ Select ]
Center $(h, k)$ : $h=$ [Select ] $k=$ [ Select ]
Length of the major axis: [ Select ]

1. Rewrite the equation
2. Complete the square
3. Analyze the ellipse

Now we can rewrite our equation in a much nicer form: $4(x-3)^2 + (y+4)^2 = 36$ .
Almost there!
To get the standard form of an ellipse equation, we need a 1 on the right side.

From the standard form, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , we can see that the center of our ellipse is $(h, k) = (3, -4)$ .
Since $36 > 9$ , the major axis is vertical (along the $y$ -axis).

The length of the major axis is $2b$ , where $b^2 = 36$ , so $b = 6$ .
Therefore, the length of the major axis is $2 \cdot 6 = 12$ .

Orientation: Vertical Center $(h, k)$ : $h = 3$ , $k = -4$ Length of the major axis: 12