shortest distance of x^2-12x+y^2-12y+68=0

To find the shortest distance from the origin (0, 0) to the curve defined by the equation \(x^2 - 12x + y^2 - 12y + 68 = 0\), we can start by rewriting the given equation in a more recognizable form, specifically that of a circle.

### Step 1: Rewrite the equation in standard form
The given equation is:

\[
x^2 - 12x + y^2 - 12y + 68 = 0
\]

We'll complete the square for both the \(x\) and \(y\) terms.

1. **Completing the square for the \(x\) terms:**

   \[
   x^2 - 12x \quad \text{can be written as} \quad (x - 6)^2 - 36
   \]

2. **Completing the square for the \(y\) terms:**

   \[
   y^2 - 12y \quad \text{can be written as} \quad (y - 6)^2 - 36
   \]

Substituting these into the original equation:

\[
(x - 6)^2 - 36 + (y - 6)^2 - 36 + 68 = 0
\]

Simplifying:

\[
(x - 6)^2 + (y - 6)^2 - 72 + 68 = 0
\]

\[
(x - 6)^2 + (y - 6)^2 - 4 = 0
\]

\[
(x - 6)^2 + (y - 6)^2 = 4
\]

### Step 2: Identify the circle's properties
The equation is now in the standard form of a circle:

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

Where \((h, k)\) is the center and \(r\) is the radius. From the equation:

- The center of the circle \((h, k)\) is \((6, 6)\).
- The radius \(r\) is \( \sqrt{4} = 2\).

### Step 3: Calculate the shortest distance from the origin
The shortest distance from the origin to the circle is the distance from the origin to the center of the circle minus the radius.

1. **Distance from the origin to the center \((6, 6)\):**

   \[
   \text{Distance} = \sqrt{(6 - 0)^2 + (6 - 0)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
   \]

2. **Shortest distance from the origin to the circle:**

   \[
   \text{Shortest Distance} = 6\sqrt{2} - 2
   \]

So, the shortest distance from the origin to the curve \(x^2 - 12x + y^2 - 12y + 68 = 0\) is \(6\sqrt{2} - 2\).

Would you like more details on any part of the solution or have any questions? Here are some related questions to consider:

1. How do you complete the square for other types of conic sections?
2. What is the geometric interpretation of completing the square?
3. How do you find the distance between two points in a coordinate plane?
4. What is the general form of the equation of a circle?
5. How do you find the intersection points between a line and a circle?

**Tip:** Completing the square is a powerful technique for transforming quadratic equations into a more recognizable form, such as the equation of a circle or a parabola.

Learn how to find the shortest distance from the origin to the circle defined by the equation x^2 - 12x + y^2 - 12y + 68 = 0. This problem involves completing the square for circle equations and applying the distance formula. Suitable for advanced high school students and beyond.

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