Find solutions to your problems with the help of IDNLearn.com's expert community. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
Sagot :
To analyze the given circle's equation [tex]\( x^2 + y^2 + 4x - 6y - 36 = 0 \)[/tex] and check the given statements, let's convert it to its standard form step-by-step.
### Step 1: Rearrange the equation
Starting with the given equation:
[tex]\[ x^2 + y^2 + 4x - 6y - 36 = 0 \][/tex]
Move the constant term to the other side:
[tex]\[ x^2 + y^2 + 4x - 6y = 36 \][/tex]
### Step 2: Complete the square
#### For the [tex]\(x\)[/tex]-terms:
Consider the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 + 4x \][/tex]
To complete the square, add and subtract the necessary constant:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
Here, we added and subtracted 4. Thus, transforming it to:
[tex]\[ (x + 2)^2 - 4 \][/tex]
#### For the [tex]\(y\)[/tex]-terms:
Consider the terms involving [tex]\(y\)[/tex]:
[tex]\[ y^2 - 6y \][/tex]
To complete the square, add and subtract the necessary constant:
[tex]\[ y^2 - 6y = (y - 3)^2 - 9 \][/tex]
Here, we added and subtracted 9. Thus, transforming it to:
[tex]\[ (y - 3)^2 - 9 \][/tex]
### Step 3: Rewrite the equation with completed squares
Substitute back the completed squares into the equation:
[tex]\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 36 \][/tex]
Combine the constants on the left-hand side:
[tex]\[ (x + 2)^2 + (y - 3)^2 - 13 = 36 \][/tex]
[tex]\[ (x + 2)^2 + (y - 3)^2 = 49 \][/tex]
### Step 4: Determine the center and radius
The equation is now in standard form:
[tex]\[ (x + 2)^2 + (y - 3)^2 = 49 \][/tex]
From this, we can see that:
- The center of the circle ([tex]\(h, k\)[/tex]) is at [tex]\((-2, 3)\)[/tex].
- The radius [tex]\(r\)[/tex] of the circle is [tex]\(\sqrt{49} = 7\)[/tex].
### Step 5: Evaluate the statements
Based on our conversion, we can evaluate the statements:
1. To begin converting the equation to standard form, subtract 36 from both sides.
- This statement is true because we moved the constant term to the other side to start the process.
2. To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides.
- This statement is true because to complete the square for [tex]\(x\)[/tex]-terms [tex]\((x + 2)^2\)[/tex], we added 4 inside the grouping.
3. The center of the circle is at [tex]\((-2, 3)\)[/tex].
- This statement is true as determined by the standard form [tex]\((x + 2)^2 + (y - 3)^2 = 49\)[/tex].
4. The center of the circle is at [tex]\((4, -6)\)[/tex].
- This statement is false; the center is [tex]\((-2, 3)\)[/tex].
5. The radius of the circle is 6 units.
- This statement is false; the radius is 7 units because [tex]\(\sqrt{49} = 7\)[/tex].
6. The radius of the circle is 49 units.
- This statement is true in the sense that the radius squared is 49, but actually, the radius length is [tex]\(\sqrt{49} = 7\)[/tex], so for practical purposes, this statement would be considered misleading or false.
Thus, the true statements are:
- To begin converting the equation to standard form, subtract 36 from both sides.
- To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides.
- The center of the circle is at [tex]\((-2, 3)\)[/tex].
### Step 1: Rearrange the equation
Starting with the given equation:
[tex]\[ x^2 + y^2 + 4x - 6y - 36 = 0 \][/tex]
Move the constant term to the other side:
[tex]\[ x^2 + y^2 + 4x - 6y = 36 \][/tex]
### Step 2: Complete the square
#### For the [tex]\(x\)[/tex]-terms:
Consider the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 + 4x \][/tex]
To complete the square, add and subtract the necessary constant:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
Here, we added and subtracted 4. Thus, transforming it to:
[tex]\[ (x + 2)^2 - 4 \][/tex]
#### For the [tex]\(y\)[/tex]-terms:
Consider the terms involving [tex]\(y\)[/tex]:
[tex]\[ y^2 - 6y \][/tex]
To complete the square, add and subtract the necessary constant:
[tex]\[ y^2 - 6y = (y - 3)^2 - 9 \][/tex]
Here, we added and subtracted 9. Thus, transforming it to:
[tex]\[ (y - 3)^2 - 9 \][/tex]
### Step 3: Rewrite the equation with completed squares
Substitute back the completed squares into the equation:
[tex]\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 36 \][/tex]
Combine the constants on the left-hand side:
[tex]\[ (x + 2)^2 + (y - 3)^2 - 13 = 36 \][/tex]
[tex]\[ (x + 2)^2 + (y - 3)^2 = 49 \][/tex]
### Step 4: Determine the center and radius
The equation is now in standard form:
[tex]\[ (x + 2)^2 + (y - 3)^2 = 49 \][/tex]
From this, we can see that:
- The center of the circle ([tex]\(h, k\)[/tex]) is at [tex]\((-2, 3)\)[/tex].
- The radius [tex]\(r\)[/tex] of the circle is [tex]\(\sqrt{49} = 7\)[/tex].
### Step 5: Evaluate the statements
Based on our conversion, we can evaluate the statements:
1. To begin converting the equation to standard form, subtract 36 from both sides.
- This statement is true because we moved the constant term to the other side to start the process.
2. To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides.
- This statement is true because to complete the square for [tex]\(x\)[/tex]-terms [tex]\((x + 2)^2\)[/tex], we added 4 inside the grouping.
3. The center of the circle is at [tex]\((-2, 3)\)[/tex].
- This statement is true as determined by the standard form [tex]\((x + 2)^2 + (y - 3)^2 = 49\)[/tex].
4. The center of the circle is at [tex]\((4, -6)\)[/tex].
- This statement is false; the center is [tex]\((-2, 3)\)[/tex].
5. The radius of the circle is 6 units.
- This statement is false; the radius is 7 units because [tex]\(\sqrt{49} = 7\)[/tex].
6. The radius of the circle is 49 units.
- This statement is true in the sense that the radius squared is 49, but actually, the radius length is [tex]\(\sqrt{49} = 7\)[/tex], so for practical purposes, this statement would be considered misleading or false.
Thus, the true statements are:
- To begin converting the equation to standard form, subtract 36 from both sides.
- To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides.
- The center of the circle is at [tex]\((-2, 3)\)[/tex].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.
