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If the circle is tangent to the line [tex]\(Ax + By + C = 0\)[/tex],

General Form of The Circle Equation:
The equation [tex]\((x-a)^2+(y-b)^2=r^2\)[/tex] can be rewritten in the general form:
[tex]\[x^2 + y^2 + Ax + By + C = 0\][/tex]

To find the center and radius of the circle:
- Center: [tex]\(\left(-\frac{A}{2}, -\frac{B}{2}\right)\)[/tex]
- Radius: [tex]\(r = \sqrt{\left(-\frac{A}{2}\right)^2 + \left(-\frac{B}{2}\right)^2 - C}\)[/tex]


Sagot :

To find the center and radius of the circle given by the equation [tex]\(x^2 + y^2 + Ax + By + C = 0\)[/tex], we can use the following steps:

1. Identify the coefficients: The coefficients from the circle equation are [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].

2. Calculate the center coordinates:
The center of the circle can be determined using the formula for the center coordinates:
[tex]\[ \left(-\frac{A}{2}, -\frac{B}{2}\right) \][/tex]
Substitute [tex]\(A = 1\)[/tex] and [tex]\(B = 1\)[/tex] into the formula:
[tex]\[ \left(-\frac{1}{2}, -\frac{1}{2}\right) \][/tex]
Thus, the center [tex]\((h, k)\)[/tex] is [tex]\((-0.5, -0.5)\)[/tex].

3. Calculate the radius:
The radius [tex]\(r\)[/tex] of the circle can be computed using the formula:
[tex]\[ r = \sqrt{\left(-\frac{A}{2}\right)^2 + \left(-\frac{B}{2}\right)^2 - C} \][/tex]
Substitute [tex]\(A = 1\)[/tex], [tex]\(B = 1\)[/tex], and [tex]\(C = -3\)[/tex] into the formula:
[tex]\[ r = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 - (-3)} \][/tex]
Calculate the terms inside the square root:
[tex]\[ \left(-\frac{1}{2}\right)^2 = 0.25 \][/tex]
[tex]\[ \left(-\frac{1}{2}\right)^2 = 0.25 \][/tex]
[tex]\[ -(-3) = 3 \][/tex]
Combine these values:
[tex]\[ r = \sqrt{0.25 + 0.25 + 3} = \sqrt{3.5} \][/tex]

4. Determine the simplified radius value:
Evaluating [tex]\(\sqrt{3.5}\)[/tex]:
[tex]\[ r \approx 1.8708286933869707 \][/tex]

Therefore, the center and radius of the circle are:
[tex]\[ \text{Center}: (-0.5, -0.5) \][/tex]
[tex]\[ \text{Radius}: 1.8708286933869707 \][/tex]

These steps ensure we correctly find the center and radius for the given circle equation [tex]\(x^2 + y^2 + x + y - 3 = 0\)[/tex].
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