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Sagot :
To convert the given general form of the circle's equation to its standard form, follow these detailed steps:
1. Simplify the General Equation:
The general form of the equation is [tex]\( 7x^2 + 7y^2 - 28x + 42y - 35 = 0 \)[/tex]. Dividing the entire equation by 7 simplifies it to:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
2. Complete the Square for [tex]\(x\)[/tex]:
To complete the square for the [tex]\(x\)[/tex]-terms in the simplified equation [tex]\( x^2 - 4x \)[/tex]:
[tex]\[ x^2 - 4x \text{ can be rewritten as } (x - 2)^2 - 4 \][/tex]
Here, [tex]\((x - 2)^2 = x^2 - 4x + 4\)[/tex], so subtracting 4 adjusts for the added 4.
3. Complete the Square for [tex]\(y\)[/tex]:
To complete the square for the [tex]\(y\)[/tex]-terms in the simplified equation [tex]\( y^2 + 6y \)[/tex]:
[tex]\[ y^2 + 6y \text{ can be rewritten as } (y + 3)^2 - 9 \][/tex]
Here, [tex]\((y + 3)^2 = y^2 + 6y + 9\)[/tex], so subtracting 9 adjusts for the added 9.
4. Form the Standard Equation:
Substituting the completed squares back into the simplified equation:
[tex]\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 - 5 = 0 \][/tex]
Combine the constants [tex]\(-4, -9, -5\)[/tex]:
[tex]\[ -4 - 9 - 5 = -18 \][/tex]
Therefore, the equation becomes:
[tex]\[ (x - 2)^2 + (y + 3)^2 - 18 = 0 \][/tex]
Or:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
5. Identify the Center and Radius:
From the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]:
- The center [tex]\((h, k)\)[/tex] is [tex]\((2, -3)\)[/tex].
- The right side is [tex]\( r^2 = 18 \)[/tex], so the radius [tex]\( r \)[/tex] is [tex]\( \sqrt{18} \)[/tex] or [tex]\( 3\sqrt{2} \approx 4.242640687119285 \)[/tex].
Therefore:
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:
[tex]\[ (2, -3) \][/tex]
The radius of the circle is:
[tex]\[ 4.242640687119285 \text{ units} \][/tex]
1. Simplify the General Equation:
The general form of the equation is [tex]\( 7x^2 + 7y^2 - 28x + 42y - 35 = 0 \)[/tex]. Dividing the entire equation by 7 simplifies it to:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
2. Complete the Square for [tex]\(x\)[/tex]:
To complete the square for the [tex]\(x\)[/tex]-terms in the simplified equation [tex]\( x^2 - 4x \)[/tex]:
[tex]\[ x^2 - 4x \text{ can be rewritten as } (x - 2)^2 - 4 \][/tex]
Here, [tex]\((x - 2)^2 = x^2 - 4x + 4\)[/tex], so subtracting 4 adjusts for the added 4.
3. Complete the Square for [tex]\(y\)[/tex]:
To complete the square for the [tex]\(y\)[/tex]-terms in the simplified equation [tex]\( y^2 + 6y \)[/tex]:
[tex]\[ y^2 + 6y \text{ can be rewritten as } (y + 3)^2 - 9 \][/tex]
Here, [tex]\((y + 3)^2 = y^2 + 6y + 9\)[/tex], so subtracting 9 adjusts for the added 9.
4. Form the Standard Equation:
Substituting the completed squares back into the simplified equation:
[tex]\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 - 5 = 0 \][/tex]
Combine the constants [tex]\(-4, -9, -5\)[/tex]:
[tex]\[ -4 - 9 - 5 = -18 \][/tex]
Therefore, the equation becomes:
[tex]\[ (x - 2)^2 + (y + 3)^2 - 18 = 0 \][/tex]
Or:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
5. Identify the Center and Radius:
From the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]:
- The center [tex]\((h, k)\)[/tex] is [tex]\((2, -3)\)[/tex].
- The right side is [tex]\( r^2 = 18 \)[/tex], so the radius [tex]\( r \)[/tex] is [tex]\( \sqrt{18} \)[/tex] or [tex]\( 3\sqrt{2} \approx 4.242640687119285 \)[/tex].
Therefore:
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:
[tex]\[ (2, -3) \][/tex]
The radius of the circle is:
[tex]\[ 4.242640687119285 \text{ units} \][/tex]
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