Question

Find the center \((h, k)\) and radius \(r\) of each circle; \((b)\) graph each circle; \((c)\) find the intercepts, if any. $$ x^{2}+y^{2}-2 x-4 y-4=0 $$

Short answer

The center is (1, 2) and the radius is 3. Intercepts are at x = -2 and 4, y = -1 and 5.

Step-by-step solution

Rewrite the Circle Equation in Standard Form

The given circle equation is $$x^2 + y^2 - 2x - 4y - 4 = 0.$$ To rewrite it in standard form, we need to complete the square for both the x and y terms. First, rearrange the equation by grouping the x terms together and the y terms together:$$x^2 - 2x + y^2 - 4y = 4.$$

Complete the Square for the x Terms

To complete the square for the x terms, add and subtract the square of half the coefficient of x. For the term $$x^2 - 2x,$$ half of -2 is -1, and eq (-1)^2 = 1. So, we add and subtract 1:$$x^2 - 2x + 1 - 1.$$

Complete the Square for the y Terms

Similarly, complete the square for the y terms. For the term $$y^2 - 4y,$$ half of -4 is -2, and eq (-2)^2 = 4. So, we add and subtract 4:$$y^2 - 4y + 4 - 4.$$

Rewrite the Equation in Standard Form

Now, rewrite the equation by including the completed squares:$$x^2 - 2x + 1 + y^2 - 4y + 4 = 4 + 1 + 4.$$ This simplifies to:$$(x - 1)^2 + (y - 2)^2 = 9.$$ The standard form of a circle is $$(x - h)^2 + (y - k)^2 = r^2,$$ where (h, k) is the center and r is the radius.

Identify the Center and Radius

From the equation $$(x - 1)^2 + (y - 2)^2 = 9,$$ we see that the center (h, k) is \((1, 2)\) and the radius r is $$eq \frac {9}{2}.$$

Graph the Circle

To graph the circle, plot the center at (1, 2) on the coordinate plane. Then, use the radius, which is 3, to plot points at a distance of 3 units in all directions (up, down, left, right) from the center. Draw a smooth curve that passes through these points to form the circle.

Find the Intercepts

To find the x-intercepts, set y to 0 in the standard form equation: $$(x - 1)^2 + (0 - 2)^2 = 9.$$ This simplifies to $$(x - 1)^2 + 4 = 9.$$ Solving for x:$$(x - 1)^2 = 5,$$x - 1 = eq \frac {eq 5}{2},$$$$eqthe \(x\)-intercepts are $$eq - 2 >< 1.$$For y-intercepts, set x to 0 in the standard form equation: $$(0 - 1)^2 + (y - 2)^2 = 9.$$ This simplifies to $$1 + (y - 2)^2 = 9.$$ Solving for y:$$(y - 2)^2 = 8,$$$$eq \frac {8}{2},$$$$The \(y\)-intercepts are $$eq - 3 >< 2$$.

Key concepts

Completing the Square

One core step in solving circle equations is 'Completing the Square'. This technique transforms the given quadratic terms into a perfect square trinomial. For the equation \(x^2 + y^2 - 2x - 4y - 4 = 0\), you group the x and y terms separately and adjust them:

For the x terms: \(x^2 - 2x\), take half of the coefficient of x (which is -2), square it to get 1, and add/subtract it:
\(x^2 - 2x + 1 - 1\).

For the y terms: \(y^2 - 4y\), take half of the coefficient of y (which is -4), square it to get 4, and add/subtract it:
\(y^2 - 4y + 4 - 4\).

This step changes the original equation into a format that can be easily converted into the standard form of a circle.

Standard Form of a Circle

Rewriting the equation in the form \((x - h)^2 + (y - k)^2 = r^2\) helps identify the circle's center and radius. From the previous steps, we formed the equation:

\((x - 1)^2 + (y - 2)^2 = 9\).

Here, the center (h, k) is \((1, 2)\) and the radius is the square root of 9, which is 3. This standard form is crucial for understanding the circle's geometric properties and for graphing.

Graphing Circles

Once the equation is in standard form, graphing the circle becomes straightforward. Start by plotting the center at \((1, 2)\) on the coordinate plane. Next, use the radius of 3 to mark points that are 3 units away from the center in all four cardinal directions (up, down, left, right). Connect these points with a smooth, round curve to complete the circle.

Intercepts

Finding intercepts involves setting either x or y to zero and solving the resulting equation:

For x-intercepts, set \(y = 0\):
\((x - 1)^2 + 2^2 = 9\)
This simplifies to \((x - 1)^2 = 5\), yielding \(x = 1 \,\pm\, \sqrt{5}\), giving intercepts\( (1 + \sqrt{5}, 0) \) and \((1 - \sqrt{5}, 0) \).

For y-intercepts, set \(x = 0\):
\(1 + (y - 2)^2 = 9\)
This simplifies to \((y - 2)^2 = 8\), yielding \(y = 2 \,\pm\, \sqrt{8}\), hence intercepts \((0, 2 + \sqrt{8})\) and \((0, 2 - \sqrt{8})\).

Calculating intercepts helps in precisely graphing the circle and understanding where it cuts through the x and y axes.