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What is the radius of the circle given by the equation below?

[tex]\[ x^2 + 12x + y^2 + 10y = -36 \][/tex]

A. 5
B. 10
C. 2.5
D. 25

Sagot :

To find the radius of the circle given by the equation [tex]\( x^2 + 12x + y^2 + 10y = -36 \)[/tex], we need to rewrite the equation in the standard form of a circle's equation, which is:

[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

Here is the step-by-step process to convert the given equation to the standard form:

1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:

[tex]\[ x^2 + 12x + y^2 + 10y = -36 \][/tex]

2. Complete the square for the [tex]\(x\)[/tex]-terms:

- Take the coefficient of [tex]\(x\)[/tex] (which is 12), divide it by 2, and square the result:

[tex]\[ \left(\frac{12}{2}\right)^2 = 36 \][/tex]

- Add and subtract 36 inside the equation to complete the square:

[tex]\[ x^2 + 12x + 36 - 36 \][/tex]

- This can be rewritten as:

[tex]\[ (x + 6)^2 - 36 \][/tex]

3. Complete the square for the [tex]\(y\)[/tex]-terms:

- Take the coefficient of [tex]\(y\)[/tex] (which is 10), divide it by 2, and square the result:

[tex]\[ \left(\frac{10}{2}\right)^2 = 25 \][/tex]

- Add and subtract 25 inside the equation to complete the square:

[tex]\[ y^2 + 10y + 25 - 25 \][/tex]

- This can be rewritten as:

[tex]\[ (y + 5)^2 - 25 \][/tex]

4. Substitute these completed squares into the original equation:

[tex]\[ (x + 6)^2 - 36 + (y + 5)^2 - 25 = -36 \][/tex]

5. Combine like terms:

[tex]\[ (x + 6)^2 + (y + 5)^2 - 61 = -36 \][/tex]

[tex]\[ (x + 6)^2 + (y + 5)^2 = -36 + 61 \][/tex]

[tex]\[ (x + 6)^2 + (y + 5)^2 = 25 \][/tex]

6. Compare this result with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]:

- Here [tex]\(r^2 = 25\)[/tex], therefore, [tex]\(r = \sqrt{25} = 5\)[/tex]

So, the radius of the circle is:

[tex]\[ \boxed{5} \][/tex]