Determine if the equation represents a circle, a point, or has no graph: 𝑥
ଶ + 𝑦
ଶ − 6𝑥 − 10𝑦 + 34 = 0

We are given the equation:

\[
x^2 + y^2 - 6x - 10y + 34 = 0
\]

To determine whether this equation represents a circle, a point, or has no graph, we can rewrite it in a more familiar form by completing the square.

### Step 1: Rearrange the equation

Group the \(x\)-terms and the \(y\)-terms together:

\[
(x^2 - 6x) + (y^2 - 10y) = -34
\]

### Step 2: Complete the square

Now, we will complete the square for both \(x\) and \(y\).

- For the \(x^2 - 6x\) term: Take half of \(-6\), square it, and add it inside the parentheses:
  
  \[
  \frac{-6}{2} = -3, \quad (-3)^2 = 9
  \]

  So, add 9 inside the parentheses for \(x\).

- For the \(y^2 - 10y\) term: Take half of \(-10\), square it, and add it inside the parentheses:
  
  \[
  \frac{-10}{2} = -5, \quad (-5)^2 = 25
  \]

  So, add 25 inside the parentheses for \(y\).

### Step 3: Adjust the equation

When completing the square, we added 9 and 25 to the left-hand side, so we need to add them to the right-hand side as well:

\[
(x^2 - 6x + 9) + (y^2 - 10y + 25) = -34 + 9 + 25
\]

Simplifying the right-hand side:

\[
(x - 3)^2 + (y - 5)^2 = 0
\]

### Step 4: Analyze the equation

The equation \((x - 3)^2 + (y - 5)^2 = 0\) represents a circle with center \((3, 5)\) and radius 0. This means the graph is a **single point** at \((3, 5)\).

### Conclusion:

The equation represents a **point**, not a full circle.

---

Would you like more details or have any other questions?

Here are five related questions for further exploration:
1. How do you complete the square in general for any quadratic expression?
2. What does it mean when the radius of a circle is zero?
3. How do you find the center and radius of a circle from its general equation?
4. Can you explain how the discriminant applies to conic sections?
5. How would the equation change if the right-hand side were not zero?

**Tip**: When dealing with conic sections, completing the square is a powerful method to rewrite and identify their standard forms.

Determine if the equation represents a circle, a point, or has no graph: x^2 + y^2 − 6x − 10y + 34 = 0

Explore the solution to the equation x^2 + y^2 − 6x − 10y + 34 = 0, including completing the square and determining whether the equation represents a circle, a point, or has no graph. The problem is explained step by step, covering algebraic concepts and geometry. Suitable for high school students in Grades 10-12.

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