Let's start by understanding the given equation and finding the center and radius of the original circle. The equation of the given circle is:

$2x^2 + 2y^2 + 4x - 2y + 1 = 0$

Step 1: Simplify the equation

Divide the entire equation by 2 to simplify it:

$x^2 + y^2 + 2x - y + \frac{1}{2} = 0$

Step 2: Complete the square

To write this equation in standard form $(x - h)^2 + (y - k)^2 = r^2$, we need to complete the square for both $x$ and $y$.

1. For the $x$-terms: $x^2 + 2x$

   • Take half of 2 (which is 1), and square it (giving 1).
   • So, $x^2 + 2x = (x + 1)^2 - 1$.

2. For the $y$-terms: $y^2 - y$

   • Take half of -1 (which is -1/2), and square it (giving 1/4).
   • So, $y^2 - y = \left( y - \frac{1}{2} \right)^2 - \frac{1}{4}$.

Step 3: Substitute back into the equation

Substitute the completed squares into the original equation:

$(x + 1)^2 - 1 + \left( y - \frac{1}{2} \right)^2 - \frac{1}{4} + \frac{1}{2} = 0$

Simplify:

$(x + 1)^2 + \left( y - \frac{1}{2} \right)^2 - \frac{3}{4} = 0$

$(x + 1)^2 + \left( y - \frac{1}{2} \right)^2 = \frac{3}{4}$

This is the standard form of the given circle with center $(-1, \frac{1}{2})$ and radius $\frac{\sqrt{3}}{2}$.

Step 4: Equation of the concentric circle

Since the new circle is concentric with the original circle, it shares the same center $(-1, \frac{1}{2})$. The equation of a circle is given by:

$(x - h)^2 + (y - k)^2 = r^2$

where $(h, k)$ is the center and $r$ is the radius. We now know the center is $(-1, \frac{1}{2})$, but we need to find the radius of the new circle. We are told that the new circle passes through the point $(4, -2)$.

Step 5: Find the radius of the new circle

To find the radius, use the distance formula to calculate the distance from the center $(-1, \frac{1}{2})$ to the point $(4, -2)$:

$r = \sqrt{(4 - (-1))^2 + \left(-2 - \frac{1}{2}\right)^2}$ $r = \sqrt{(4 + 1)^2 + \left(-2 - \frac{1}{2}\right)^2}$ $r = \sqrt{5^2 + \left(-\frac{5}{2}\right)^2}$ $r = \sqrt{25 + \frac{25}{4}}$ $r = \sqrt{\frac{100}{4} + \frac{25}{4}}$ $r = \sqrt{\frac{125}{4}} = \frac{\sqrt{125}}{2} = \frac{5\sqrt{5}}{2}$

Step 6: Write the equation of the new circle

Now that we know the center $(-1, \frac{1}{2})$ and the radius $\frac{5\sqrt{5}}{2}$, the equation of the new circle is:

$(x + 1)^2 + \left( y - \frac{1}{2} \right)^2 = \left( \frac{5\sqrt{5}}{2} \right)^2$

Simplify the radius squared:

$(x + 1)^2 + \left( y - \frac{1}{2} \right)^2 = \frac{125}{4}$

Final Answer:

The equation of the new circle is:

$(x + 1)^2 + \left( y - \frac{1}{2} \right)^2 = \frac{125}{4}$

---

Would you like a more detailed explanation of any step or have any questions?

Here are 5 related questions for you:

1. How do you complete the square for more complicated equations involving other terms like $x^2$ and $y^2$?
2. Can the method of completing the square be used for other conic sections, such as ellipses or hyperbolas?
3. How would the equation of the circle change if the center moved to another point, say $(2, 3)$?
4. How do you derive the distance formula, and what is its geometric significance in circle equations?
5. What happens to the equation of the circle if we scale the entire equation by a constant factor?

Tip: When completing the square, always remember to add and subtract the same value to maintain the equality.