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Sagot :
To convert the given general form of the equation of a circle [tex]\(7x^2 + 7y^2 - 28x + 42y - 35 = 0\)[/tex] to its standard form, we need to follow a few steps:
1. Simplify the equation:
First, divide the entire equation by 7 to make the coefficients of [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] equal to 1:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
2. Rearrange and group terms:
Rearrange and group the [tex]\(x\)[/tex] terms together and the [tex]\(y\)[/tex] terms together:
[tex]\[ (x^2 - 4x) + (y^2 + 6y) = 5 \][/tex]
3. Complete the square:
We complete the square separately for the [tex]\(x\)[/tex] terms and the [tex]\(y\)[/tex] terms.
For [tex]\(x\)[/tex]-terms:
[tex]\[ x^2 - 4x \quad \Rightarrow \quad (x - 2)^2 - 4 \][/tex]
For [tex]\(y\)[/tex]-terms:
[tex]\[ y^2 + 6y \quad \Rightarrow \quad (y + 3)^2 - 9 \][/tex]
4. Substitute back into the equation:
Substitute these completed squares back into the rearranged equation:
[tex]\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 = 5 \][/tex]
5. Simplify:
Combine like terms and simplify:
[tex]\[ (x - 2)^2 + (y + 3)^2 - 13 = 5 \quad \Rightarrow \quad (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
Thus, the equation of the circle in standard form is:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
6. Identify the center and radius:
The standard form of a circle's 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 the equation [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex], we can see that the center of the circle is:
[tex]\[ (h, k) = (2, -3) \][/tex]
And the radius [tex]\(r\)[/tex] is the square root of 18:
[tex]\[ r = \sqrt{18} \approx 4.242640687119285 \][/tex]
Therefore:
- The standard form of the equation of this circle 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 approximately [tex]\(4.242640687119285\)[/tex] units.
So the correct answers are:
- 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].
- Its radius is approximately [tex]\(4.242640687119285\)[/tex] units.
1. Simplify the equation:
First, divide the entire equation by 7 to make the coefficients of [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] equal to 1:
[tex]\[ x^2 + y^2 - 4x + 6y - 5 = 0 \][/tex]
2. Rearrange and group terms:
Rearrange and group the [tex]\(x\)[/tex] terms together and the [tex]\(y\)[/tex] terms together:
[tex]\[ (x^2 - 4x) + (y^2 + 6y) = 5 \][/tex]
3. Complete the square:
We complete the square separately for the [tex]\(x\)[/tex] terms and the [tex]\(y\)[/tex] terms.
For [tex]\(x\)[/tex]-terms:
[tex]\[ x^2 - 4x \quad \Rightarrow \quad (x - 2)^2 - 4 \][/tex]
For [tex]\(y\)[/tex]-terms:
[tex]\[ y^2 + 6y \quad \Rightarrow \quad (y + 3)^2 - 9 \][/tex]
4. Substitute back into the equation:
Substitute these completed squares back into the rearranged equation:
[tex]\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 = 5 \][/tex]
5. Simplify:
Combine like terms and simplify:
[tex]\[ (x - 2)^2 + (y + 3)^2 - 13 = 5 \quad \Rightarrow \quad (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
Thus, the equation of the circle in standard form is:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 18 \][/tex]
6. Identify the center and radius:
The standard form of a circle's 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 the equation [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex], we can see that the center of the circle is:
[tex]\[ (h, k) = (2, -3) \][/tex]
And the radius [tex]\(r\)[/tex] is the square root of 18:
[tex]\[ r = \sqrt{18} \approx 4.242640687119285 \][/tex]
Therefore:
- The standard form of the equation of this circle 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 approximately [tex]\(4.242640687119285\)[/tex] units.
So the correct answers are:
- 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].
- Its radius is approximately [tex]\(4.242640687119285\)[/tex] units.
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