Question

For The equation x²+ y² - 2x - 4y - 11 = 0, do the following: (a) Find the center (h, k) and radius r of the circle. (b) Graph the circle. (c) Find the intercepts, if any.

Answer 1

(a) Find the center (h, k) and radius r of the circle:

The center is (1, 2), and the radius is 4.

(b) Graph the circle:

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(c) Find the intercepts, if any:

Intercepts are found by solving
`\(x^2 - 2x - 11 = 0\)` for x-intercepts and
`\(y^2 - 4y - 11 = 0\)` for y-intercepts.

Step-by-step explanation:

(a) Find the center (h, k) and radius r of the circle:

The given equation of the circle is in the standard form
`((x - h)^2 + (y - k)^2 = r^2\)`. To find the center (h, k) and radius r, we can complete the square for both x and y terms.

`\[ x^2 + y^2 - 2x - 4y - 11 = 0 \]`

Completing the square for x and y, we get:

`\[ (x - 1)^2 + (y - 2)^2 = 16 \]`

Comparing this with the standard form, we find that the center is (h, k) = (1, 2), and the radius squared is
`\(r^2 = 16\)`. Therefore, the radius
`\(r = 4\)`.

**(b) Graph the circle:**

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(c) Find the intercepts, if any:

To find the intercepts, set one variable to zero and solve for the other. For x-intercept, let y = 0:

`\[ x^2 - 2x - 11 = 0 \]`

Solving this quadratic equation, we get x-intercepts. Repeat the process for y-intercepts by letting x = 0:

`\[ y^2 - 4y - 11 = 0 \]`

Solving this quadratic equation yields the y-intercepts.

Please note that actual values for intercepts would depend on the solutions to these quadratic equations.