The problem asks us to find the value of $x^2 + y^2$ given the equation of a circle in general form: $x^2 + y^2 - 12x + 18y + 117 = 0$.

Solving the Circle Equation

To find the values of $x$ and $y$, we first need to convert the given general form of the circle equation into its standard form, which is $(x-h)^2 + (y-k)^2 = r^2$. We achieve this by completing the square for the $x$ and $y$ terms.

1. Group terms: Rearrange the equation to group the $x$ terms and $y$ terms together: $$ (x^2 - 12x) + (y^2 + 18y) + 117 = 0 $$
2. Complete the square for x: Take half of the coefficient of the $x$ term (which is -12), square it ($(\frac{-12}{2})^2 = (-6)^2 = 36$), and add and subtract it within the $x$ group: $$ (x^2 - 12x + 36) - 36 $$ This simplifies to: $$ (x - 6)^2 - 36 $$
3. Complete the square for y: Take half of the coefficient of the $y$ term (which is 18), square it ($(\frac{18}{2})^2 = (9)^2 = 81$), and add and subtract it within the $y$ group: $$ (y^2 + 18y + 81) - 81 $$ This simplifies to: $$ (y + 9)^2 - 81 $$
4. Substitute back and simplify: Substitute the completed square forms back into the grouped equation: $$ ((x - 6)^2 - 36) + ((y + 9)^2 - 81) + 117 = 0 $$ Now, simplify the constants: $$ (x - 6)^2 + (y + 9)^2 - 36 - 81 + 117 = 0 $$ $$ (x - 6)^2 + (y + 9)^2 - 117 + 117 = 0 $$ $$ (x - 6)^2 + (y + 9)^2 = 0 $$

Determining the Values of x and y

The equation $(x - 6)^2 + (y + 9)^2 = 0$ represents the sum of two squared terms equaling zero. Since the square of any real number is non-negative (greater than or equal to zero), the only way for the sum of two squares to be zero is if each individual term is zero.

• Set the $x$ term to zero: $$ (x - 6)^2 = 0 $$ $$ x - 6 = 0 $$ $$ x = 6 $$
• Set the $y$ term to zero: $$ (y + 9)^2 = 0 $$ $$ y + 9 = 0 $$ $$ y = -9 $$

So, the only point that satisfies the given circle equation is $(6, -9)$. This is a special case where the circle degenerates to a single point.

Calculating the Value of x² + y²

Now that we have found the specific values for $x$ and $y$, we can calculate $x^2 + y^2$.

• Substitute $x = 6$ and $y = -9$ into the expression $x^2 + y^2$: $$ x^2 + y^2 = (6)^2 + (-9)^2 $$
• Perform the squaring: $$ x^2 + y^2 = 36 + 81 $$
• Add the results: $$ x^2 + y^2 = 117 $$

Therefore, the value of $x^2 + y^2$ is 117.