Math / Geometry

QuestionDecide whether or not the equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. $x^{2}+y^{2}+4 x-8 y+24=0$
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The graph of the equation is a point. B. The graph of the equation is a line. C. The graph of the equation is a circle with center $\square$ . (Type an ordered pair.) The radius of the circle is $\square$ . D. The graph is nonexistent.

Studdy Solution

STEP 1

What is this asking? Determine if the given equation represents a circle, and if so, find its center and radius. Watch out! Don't forget to complete the square for both $x$ and $y$ terms to identify the circle's properties.

STEP 2

1. Rearrange the equation
2. Complete the square for $x$
3. Complete the square for $y$
4. Identify the circle's center and radius

STEP 3

Let's start by grouping the $x$ terms and the $y$ terms together.
The equation is:
$x^{2} + 4x + y^{2} - 8y + 24 = 0$

STEP 4

Move the constant term to the other side of the equation:
$x^{2} + 4x + y^{2} - 8y = -24$

STEP 5

To complete the square for the $x$ terms, take the coefficient of $x$ , which is 4, divide it by 2, and square it:
$\left(\frac{4}{2}\right)^2 = 4$

STEP 6

Add and subtract this value inside the equation to maintain balance:
$x^{2} + 4x + 4 - 4 + y^{2} - 8y = -24$

STEP 7

Now, let's complete the square for the $y$ terms.
Take the coefficient of $y$ , which is -8, divide it by 2, and square it:
$\left(\frac{-8}{2}\right)^2 = 16$

STEP 8

Add and subtract this value inside the equation:
$x^{2} + 4x + 4 + y^{2} - 8y + 16 - 4 - 16 = -24$

STEP 9

Rearrange the equation to show the completed squares:
$(x + 2)^2 + (y - 4)^2 = -24 + 4 + 16$

STEP 10

Calculate the right-hand side:
$(x + 2)^2 + (y - 4)^2 = -24 + 20 = -4$

STEP 11

Since the right-hand side is negative, this equation does not represent a circle.
A circle's equation requires a positive value on the right side.

STEP 12

The graph is nonexistent because the equation does not satisfy the conditions for a circle.

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Studdy Solution

STEP 1

What is this asking? Determine if the given equation represents a circle, and if so, find its center and radius. Watch out! Don't forget to complete the square for both $x$ and $y$ terms to identify the circle's properties.

STEP 2

1. Rearrange the equation
2. Complete the square for $x$
3. Complete the square for $y$
4. Identify the circle's center and radius

STEP 3

Let's start by grouping the $x$ terms and the $y$ terms together.
The equation is:
$x^{2} + 4x + y^{2} - 8y + 24 = 0$

STEP 4

Move the constant term to the other side of the equation:
$x^{2} + 4x + y^{2} - 8y = -24$

STEP 5

To complete the square for the $x$ terms, take the coefficient of $x$ , which is 4, divide it by 2, and square it:
$\left(\frac{4}{2}\right)^2 = 4$

STEP 6

Add and subtract this value inside the equation to maintain balance:
$x^{2} + 4x + 4 - 4 + y^{2} - 8y = -24$

STEP 7

Now, let's complete the square for the $y$ terms.
Take the coefficient of $y$ , which is -8, divide it by 2, and square it:
$\left(\frac{-8}{2}\right)^2 = 16$

STEP 8

Add and subtract this value inside the equation:
$x^{2} + 4x + 4 + y^{2} - 8y + 16 - 4 - 16 = -24$

STEP 9

Rearrange the equation to show the completed squares:
$(x + 2)^2 + (y - 4)^2 = -24 + 4 + 16$

STEP 10

Calculate the right-hand side:
$(x + 2)^2 + (y - 4)^2 = -24 + 20 = -4$

STEP 11

Since the right-hand side is negative, this equation does not represent a circle.
A circle's equation requires a positive value on the right side.

STEP 12

The graph is nonexistent because the equation does not satisfy the conditions for a circle.

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Studdy Solution

STEP 1

What is this asking? Determine if the given equation represents a circle, and if so, find its **center** and **radius**. Watch out! Don't forget to **complete the square** for both xxx and yyy terms to identify the circle's properties.

STEP 2

1. Rearrange the equation2. Complete the square for xxx3. Complete the square for yyy4. Identify the circle's center and radius

STEP 3

Let's start by grouping the xxx terms and the yyy terms together.The equation is:x2+4x+y2−8y+24=0 x^{2} + 4x + y^{2} - 8y + 24 = 0 x2+4x+y2−8y+24=0

STEP 4

Move the constant term to the other side of the equation:x2+4x+y2−8y=−24 x^{2} + 4x + y^{2} - 8y = -24 x2+4x+y2−8y=−24

STEP 5

To complete the square for the xxx terms, take the coefficient of xxx, which is **4**, divide it by 2, and square it:(42)2=4 \left(\frac{4}{2}\right)^2 = 4 (24​)2=4

STEP 6

Add and subtract this value inside the equation to maintain balance:x2+4x+4−4+y2−8y=−24 x^{2} + 4x + 4 - 4 + y^{2} - 8y = -24 x2+4x+4−4+y2−8y=−24

STEP 7

Now, let's complete the square for the yyy terms.Take the coefficient of yyy, which is **-8**, divide it by 2, and square it:(−82)2=16 \left(\frac{-8}{2}\right)^2 = 16 (2−8​)2=16

STEP 8

Add and subtract this value inside the equation:x2+4x+4+y2−8y+16−4−16=−24 x^{2} + 4x + 4 + y^{2} - 8y + 16 - 4 - 16 = -24 x2+4x+4+y2−8y+16−4−16=−24

STEP 9

Rearrange the equation to show the completed squares:(x+2)2+(y−4)2=−24+4+16 (x + 2)^2 + (y - 4)^2 = -24 + 4 + 16 (x+2)2+(y−4)2=−24+4+16

STEP 10

Calculate the right-hand side:(x+2)2+(y−4)2=−24+20=−4 (x + 2)^2 + (y - 4)^2 = -24 + 20 = -4 (x+2)2+(y−4)2=−24+20=−4

STEP 11

Since the right-hand side is negative, this equation does not represent a circle.A circle's equation requires a positive value on the right side.

STEP 12

The graph is **nonexistent** because the equation does not satisfy the conditions for a circle.

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

What is this asking? Determine if the given equation represents a circle, and if so, find its center and radius. Watch out! Don't forget to complete the square for both $x$ and $y$ terms to identify the circle's properties.

1. Rearrange the equation
2. Complete the square for $x$
3. Complete the square for $y$
4. Identify the circle's center and radius

Let's start by grouping the $x$ terms and the $y$ terms together.
The equation is:
$x^{2} + 4x + y^{2} - 8y + 24 = 0$

Move the constant term to the other side of the equation:
$x^{2} + 4x + y^{2} - 8y = -24$

To complete the square for the $x$ terms, take the coefficient of $x$ , which is 4, divide it by 2, and square it:
$\left(\frac{4}{2}\right)^2 = 4$

Add and subtract this value inside the equation to maintain balance:
$x^{2} + 4x + 4 - 4 + y^{2} - 8y = -24$

Now, let's complete the square for the $y$ terms.
Take the coefficient of $y$ , which is -8, divide it by 2, and square it:
$\left(\frac{-8}{2}\right)^2 = 16$

Add and subtract this value inside the equation:
$x^{2} + 4x + 4 + y^{2} - 8y + 16 - 4 - 16 = -24$

Rearrange the equation to show the completed squares:
$(x + 2)^2 + (y - 4)^2 = -24 + 4 + 16$

Calculate the right-hand side:
$(x + 2)^2 + (y - 4)^2 = -24 + 20 = -4$

Since the right-hand side is negative, this equation does not represent a circle.
A circle's equation requires a positive value on the right side.

The graph is nonexistent because the equation does not satisfy the conditions for a circle.