Answer :
To convert the given equation of a circle from its general form to its standard form, follow these steps carefully:
### 1. Write Down the Given General Form
The given equation is:
[tex]\[ 7x^2 + 7y^2 - 28x + 42y - 35 = 0 \][/tex]
### 2. Simplify the Equation
First, divide the entire equation by 7 to make it simpler:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
### 3. Rearrange and Prepare to Complete the Square
Rearrange the terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] together:
[tex]\[ x^2 - 4x + y^2 + 6y = 5 \][/tex]
### 4. Complete the Square
Next, complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
#### For [tex]\(x\)[/tex]:
- Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-4\)[/tex], divide it by 2 to get [tex]\(-2\)[/tex], and then square it to get 4.
- Add and subtract 4 inside the equation to keep it balanced.
So,
[tex]\[ x^2 - 4x + 4 \][/tex]
#### For [tex]\(y\)[/tex]:
- Take the coefficient of [tex]\(y\)[/tex], which is 6, divide it by 2 to get 3, and then square it to get 9.
- Add and subtract 9 inside the equation to keep it balanced.
So,
[tex]\[ y^2 + 6y + 9 \][/tex]
With these modifications, the original equation becomes:
[tex]\[ (x^2 - 4x + 4) + (y^2 + 6y + 9) = 5 + 4 + 9 \][/tex]
### 5. Rewrite into Perfect Square Form
Now express the completed squares in their factorized forms:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
### 6. Identify the Center and Radius
The standard form of the circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2 \)[/tex].
From the equation [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex], we can directly identify:
- The center [tex]\((h, k)\)[/tex] is [tex]\((2, -3)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{18}\)[/tex], which simplifies to [tex]\(3\sqrt{2}\)[/tex].
So, the standard form of the equation is:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
### Conclusion
- The equation of this circle in standard form is: [tex]\((x - 2)^2 + (y + 3)^2 = 49 \)[/tex].
- The center of the circle is at the point [tex]\((2, -3)\)[/tex].
### 1. Write Down the Given General Form
The given equation is:
[tex]\[ 7x^2 + 7y^2 - 28x + 42y - 35 = 0 \][/tex]
### 2. Simplify the Equation
First, divide the entire equation by 7 to make it simpler:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
### 3. Rearrange and Prepare to Complete the Square
Rearrange the terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] together:
[tex]\[ x^2 - 4x + y^2 + 6y = 5 \][/tex]
### 4. Complete the Square
Next, complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
#### For [tex]\(x\)[/tex]:
- Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-4\)[/tex], divide it by 2 to get [tex]\(-2\)[/tex], and then square it to get 4.
- Add and subtract 4 inside the equation to keep it balanced.
So,
[tex]\[ x^2 - 4x + 4 \][/tex]
#### For [tex]\(y\)[/tex]:
- Take the coefficient of [tex]\(y\)[/tex], which is 6, divide it by 2 to get 3, and then square it to get 9.
- Add and subtract 9 inside the equation to keep it balanced.
So,
[tex]\[ y^2 + 6y + 9 \][/tex]
With these modifications, the original equation becomes:
[tex]\[ (x^2 - 4x + 4) + (y^2 + 6y + 9) = 5 + 4 + 9 \][/tex]
### 5. Rewrite into Perfect Square Form
Now express the completed squares in their factorized forms:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
### 6. Identify the Center and Radius
The standard form of the circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2 \)[/tex].
From the equation [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex], we can directly identify:
- The center [tex]\((h, k)\)[/tex] is [tex]\((2, -3)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{18}\)[/tex], which simplifies to [tex]\(3\sqrt{2}\)[/tex].
So, the standard form of the equation is:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
### Conclusion
- The equation of this circle in standard form is: [tex]\((x - 2)^2 + (y + 3)^2 = 49 \)[/tex].
- The center of the circle is at the point [tex]\((2, -3)\)[/tex].
