Find distance between (5.8) and circumference of xpower2 +ypower2 + 2x +6y +3=0

To find the distance between the point (5, 8) and the circumference (the boundary) of the circle defined by the equation \( x^2 + y^2 + 2x + 6y + 3 = 0 \), we follow these steps:

### Step 1: Convert the circle equation to standard form

The given circle equation is:
\[
x^2 + y^2 + 2x + 6y + 3 = 0
\]

We can rewrite this equation by completing the square for the \(x\) and \(y\) terms.

1. **Group the x-terms and y-terms:**
   \[
   (x^2 + 2x) + (y^2 + 6y) = -3
   \]

2. **Complete the square:**
   - For the x-term: \( x^2 + 2x \), add and subtract \(\left(\frac{2}{2}\right)^2 = 1\).
   - For the y-term: \( y^2 + 6y \), add and subtract \(\left(\frac{6}{2}\right)^2 = 9\).

   So, the equation becomes:
   \[
   (x^2 + 2x + 1) + (y^2 + 6y + 9) = -3 + 1 + 9
   \]

   Simplifying, we get:
   \[
   (x + 1)^2 + (y + 3)^2 = 7
   \]

Now, the equation is in the standard form:
\[
(x + 1)^2 + (y + 3)^2 = 7
\]

### Step 2: Identify the center and radius of the circle

From the standard form \( (x + 1)^2 + (y + 3)^2 = 7 \), we can see:
- The center of the circle is \( (-1, -3) \).
- The radius \( r \) is \( \sqrt{7} \).

### Step 3: Calculate the distance from the point (5, 8) to the center (-1, -3)

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the values:
\[
d = \sqrt{(5 - (-1))^2 + (8 - (-3))^2} = \sqrt{(5 + 1)^2 + (8 + 3)^2} = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157}
\]

### Step 4: Calculate the distance from the point (5, 8) to the circumference

The distance from the point to the circumference of the circle is the distance from the point to the center minus the radius:
\[
\text{Distance} = \sqrt{157} - \sqrt{7}
\]

So, the distance between the point \( (5, 8) \) and the circumference of the circle is \( \sqrt{157} - \sqrt{7} \).

Would you like further details or have any questions?

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**Related Questions:**
1. How do you derive the formula for the distance between a point and a line?
2. What is the significance of completing the square when dealing with circle equations?
3. Can you explain the geometric interpretation of the distance from a point to a circle's circumference?
4. What are the properties of a circle's radius?
5. How can we determine if a point lies inside, outside, or on a circle?

**Tip:** Understanding the geometry of circles is fundamental for solving problems involving distance, tangents, and angles related to circles.

Learn how to find the distance between a point (5, 8) and the circumference of the circle defined by the equation x^2 + y^2 + 2x + 6y + 3 = 0. This problem involves using circle equations and the distance formula. Suitable for high school students.

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