Answer :
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].
