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Sagot :
To find the standard form of the given equation of a circle and determine its center and radius, let's proceed step-by-step.
### 1. Rewrite the Equation in a Simplified Form:
Given the equation:
[tex]\[ 7x^2 + 7y^2 - 28x + 42y - 35 = 0 \][/tex]
First, simplify by dividing the entire equation by 7:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
### 2. Complete the Square:
To convert this equation into the standard form of a circle [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
#### Completing the Square for [tex]\(x\)[/tex]:
For the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 - 4x \][/tex]
Take half of the coefficient of [tex]\(x\)[/tex] (-4), square it, and add it inside the parentheses:
[tex]\[ \left(\frac{-4}{2}\right)^2 = (-2)^2 = 4 \][/tex]
So, we rewrite:
[tex]\[ x^2 - 4x = (x - 2)^2 - 4 \][/tex]
#### Completing the Square for [tex]\(y\)[/tex]:
For the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 + 6y \][/tex]
Take half of the coefficient of [tex]\(y\)[/tex] (6), square it, and add it inside the parentheses:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
So, we rewrite:
[tex]\[ y^2 + 6y = (y + 3)^2 - 9 \][/tex]
### 3. Rewrite the Original Equation:
Incorporate these squared terms and adjust the constants appropriately:
[tex]\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 - 5 = 0 \][/tex]
Combine the constants on the right-hand side:
[tex]\[ (x - 2)^2 + (y + 3)^2 - 18 = 0 \][/tex]
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
### 4. Identify the Center and Radius:
The standard form of the equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
From our equation:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
We identify:
- Center [tex]\((h, k) = (2, -3)\)[/tex]
- Radius [tex]\(r = \sqrt{18} = 3\sqrt{2} = 4.24\)[/tex] (approx)
### Conclusion:
- The equation of this circle in standard form is: [tex]\[(x - 2)^2 + (y + 3)^2 = 18\][/tex]
- The center of the circle is at the point: [tex]\((2, -3)\)[/tex]
- The radius of the circle is: [tex]\(4.24\)[/tex] units
So, the correctly completed sentences should be:
- The general form of the equation of a circle is [tex]\(7 x^2+7 y^2-28 x+42 y-35=0\)[/tex].
- The equation of this circle in standard form is [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex]
- The center of the circle is at the point [tex]\((2, -3)\)[/tex].
- The radius of the circle is [tex]\(4.24\)[/tex] units.
### 1. Rewrite the Equation in a Simplified Form:
Given the equation:
[tex]\[ 7x^2 + 7y^2 - 28x + 42y - 35 = 0 \][/tex]
First, simplify by dividing the entire equation by 7:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
### 2. Complete the Square:
To convert this equation into the standard form of a circle [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
#### Completing the Square for [tex]\(x\)[/tex]:
For the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 - 4x \][/tex]
Take half of the coefficient of [tex]\(x\)[/tex] (-4), square it, and add it inside the parentheses:
[tex]\[ \left(\frac{-4}{2}\right)^2 = (-2)^2 = 4 \][/tex]
So, we rewrite:
[tex]\[ x^2 - 4x = (x - 2)^2 - 4 \][/tex]
#### Completing the Square for [tex]\(y\)[/tex]:
For the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 + 6y \][/tex]
Take half of the coefficient of [tex]\(y\)[/tex] (6), square it, and add it inside the parentheses:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
So, we rewrite:
[tex]\[ y^2 + 6y = (y + 3)^2 - 9 \][/tex]
### 3. Rewrite the Original Equation:
Incorporate these squared terms and adjust the constants appropriately:
[tex]\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 - 5 = 0 \][/tex]
Combine the constants on the right-hand side:
[tex]\[ (x - 2)^2 + (y + 3)^2 - 18 = 0 \][/tex]
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
### 4. Identify the Center and Radius:
The standard form of the equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
From our equation:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
We identify:
- Center [tex]\((h, k) = (2, -3)\)[/tex]
- Radius [tex]\(r = \sqrt{18} = 3\sqrt{2} = 4.24\)[/tex] (approx)
### Conclusion:
- The equation of this circle in standard form is: [tex]\[(x - 2)^2 + (y + 3)^2 = 18\][/tex]
- The center of the circle is at the point: [tex]\((2, -3)\)[/tex]
- The radius of the circle is: [tex]\(4.24\)[/tex] units
So, the correctly completed sentences should be:
- The general form of the equation of a circle is [tex]\(7 x^2+7 y^2-28 x+42 y-35=0\)[/tex].
- The equation of this circle in standard form is [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex]
- The center of the circle is at the point [tex]\((2, -3)\)[/tex].
- The radius of the circle is [tex]\(4.24\)[/tex] units.
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