2. A student from Tanza National Comprehensive High School got curious about the circular logo just outside their school. He speculates that the logo has a diameter of 2 meters. He tells his math teacher that the equation of the circle can be 2x^2 +2y^2 -4x -4y -2 = 0 if 1 meter is treated as 1 unit. His teacher says that his constant is wrong. What should the constant be to make his answer right? Give the complete given, unknown complete solution step by step and final answer 

### Given:
- The student speculates that the diameter of the circle is 2 meters, and 1 meter is treated as 1 unit. Therefore, the radius of the circle is:
\[
r = \frac{2}{2} = 1 \text{ unit}.
\]
- The equation provided by the student is:
\[
2x^2 + 2y^2 - 4x - 4y - 2 = 0.
\]
- The teacher says that the constant term in this equation is wrong.

### Unknown:
- The correct constant value to make the equation represent the circle with radius 1 unit.

### Step-by-step solution:

1. **Rewrite the given equation:**
   Start by simplifying the equation:
   \[
   2x^2 + 2y^2 - 4x - 4y - 2 = 0.
   \]
   Factor out the common factor of 2 from the terms involving \(x\) and \(y\):
   \[
   2(x^2 + y^2 - 2x - 2y) = 2.
   \]
   Divide both sides by 2:
   \[
   x^2 + y^2 - 2x - 2y = 1.
   \]
   This is now a simpler equation of a circle, but we need to complete the square to put it into the standard form.

2. **Complete the square:**
   Group the \(x\) and \(y\) terms together:
   \[
   (x^2 - 2x) + (y^2 - 2y) = 1.
   \]
   To complete the square:
   - For \(x^2 - 2x\), add and subtract \(\left(\frac{2}{2}\right)^2 = 1\):
     \[
     (x^2 - 2x + 1) - 1 = x^2 - 2x.
     \]
   - For \(y^2 - 2y\), add and subtract \(\left(\frac{2}{2}\right)^2 = 1\):
     \[
     (y^2 - 2y + 1) - 1 = y^2 - 2y.
     \]
   Now the equation becomes:
   \[
   (x - 1)^2 - 1 + (y - 1)^2 - 1 = 1.
   \]
   Simplify:
   \[
   (x - 1)^2 + (y - 1)^2 - 2 = 1.
   \]
   Move the \(-2\) to the right-hand side:
   \[
   (x - 1)^2 + (y - 1)^2 = 3.
   \]

3. **Interpret the final equation:**
   The equation \((x - 1)^2 + (y - 1)^2 = 3\) represents a circle with:
   - **Center:** \((1, 1)\)
   - **Radius:** \(\sqrt{3}\)

   However, the circle is supposed to have a radius of 1 unit (since the diameter is 2 meters). The correct form for the equation of a circle with center \((1, 1)\) and radius 1 is:
   \[
   (x - 1)^2 + (y - 1)^2 = 1^2 = 1.
   \]

4. **Find the correct constant:**
   Comparing the simplified form from the student's equation, \((x - 1)^2 + (y - 1)^2 = 3\), with the desired equation, \((x - 1)^2 + (y - 1)^2 = 1\), the constant term is off by 2.

   Therefore, the correct constant in the original equation should be \(-6\) instead of \(-2\). This is because:
   \[
   2(x^2 + y^2 - 2x - 2y) = 6.
   \]

Thus, the correct constant should be \(6\).

### Final Answer:
The correct constant to make the student's equation represent a circle with radius 1 unit is **6**.

---

Do you need any further details on any part of this explanation? Here are 5 related questions:

1. How do you derive the standard form of a circle equation from its general form?
2. What are the steps for completing the square in an equation?
3. How can you verify that an equation represents a circle?
4. How do you calculate the radius of a circle from its equation?
5. Why is completing the square important for solving circle-related problems?

**Tip:** When completing the square, always double-check your signs and constants to ensure the equation reflects the correct circle properties.

A student from Tanza National Comprehensive High School got curious about the circular logo just outside their school. He speculates that the logo has a diameter of 2 meters. He tells his math teacher that the equation of the circle can be 2x^2 +2y^2 -4x -4y -2 = 0 if 1 meter is treated as 1 unit. His teacher says that his constant is wrong. What should the constant be to make his answer right?

Discover how to correct the constant term in the equation of a circle provided by a curious student. This problem involves completing the square and understanding the equation of a circle, perfect for students in Grades 9-12. Follow a detailed solution to find the right constant for a circle with a radius of 1 unit.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation