Key Concepts and Formulas

• General Equation of a Circle: The general equation of a circle is given by $x^2 + y^2 + 2gx + 2fy + c = 0$, where $(-g, -f)$ is the center and $\sqrt{g^2 + f^2 - c}$ is the radius.
• Condition for Orthogonal Intersection of Two Circles: Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ intersect orthogonally if $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.
• Locus: The locus of a point is the path traced by the point as it moves according to certain given conditions.

Step-by-Step Solution

Step 1: Define the Variable Circle

Let the equation of the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. The center of this circle is $(-g, -f)$. Our goal is to find the locus of this center.

Explanation: We start by assuming the general form of the circle equation. The coefficients $g$, $f$, and $c$ are parameters that define the circle. We want to find a relationship between $g$ and $f$ (or rather, $-g$ and $-f$, the coordinates of the center) based on the given conditions.

Step 2: Apply the Condition that the Circle Passes Through (a, b)

Since the circle passes through the point $(a, b)$, we substitute $x = a$ and $y = b$ into the equation of the circle: $a^2 + b^2 + 2ga + 2fb + c = 0$

Explanation: If a point lies on the circle, its coordinates must satisfy the circle's equation. Substituting $(a, b)$ into the equation gives us a relationship between $a$, $b$, $g$, $f$, and $c$.

Step 3: Apply the Orthogonality Condition

The given circle is $x^2 + y^2 = 4$, which can be written as $x^2 + y^2 - 4 = 0$. Comparing this with the general form, we have $g_2 = 0$, $f_2 = 0$, and $c_2 = -4$. The other circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, so $g_1 = g$, $f_1 = f$, and $c_1 = c$. Using the condition for orthogonal intersection, $2g_1g_2 + 2f_1f_2 = c_1 + c_2$, we get: $2(g)(0) + 2(f)(0) = c + (-4)$ $0 = c - 4$ $c = 4$

Explanation: We are given that the two circles intersect orthogonally. We identify the coefficients from the two circle equations and apply the orthogonality condition. This gives us the value of $c$.

Step 4: Substitute the value of c back into the equation from Step 2

Substitute $c = 4$ into the equation $a^2 + b^2 + 2ga + 2fb + c = 0$: $a^2 + b^2 + 2ga + 2fb + 4 = 0$

Explanation: We now have an equation relating $a$, $b$, $g$, and $f$.

Step 5: Find the Locus of the Center

We want to find the locus of the center $(-g, -f)$. Let $x = -g$ and $y = -f$. Thus, $g = -x$ and $f = -y$. Substitute these into the equation: $a^2 + b^2 + 2(-x)a + 2(-y)b + 4 = 0$ $a^2 + b^2 - 2ax - 2by + 4 = 0$ $2ax + 2by - (a^2 + b^2 + 4) = 0$

Explanation: The locus is a relationship between the coordinates of the center. We replace $-g$ with $x$ and $-f$ with $y$ to get the equation of the locus in terms of $x$ and $y$.

Common Mistakes & Tips

• Sign Errors: Be very careful with signs, especially when substituting $x = -g$ and $y = -f$.
• Orthogonality Condition: Remember the correct formula for orthogonal intersection.
• Locus Definition: Understand that the locus is an equation that relates the coordinates of a moving point (in this case, the center of the circle).

Summary

We started with the general equation of a circle and used the given conditions (passing through a point and orthogonal intersection) to find a relationship between the parameters of the circle. Then, by substituting the coordinates of the center $(-g, -f)$ with $(x, y)$, we obtained the locus of the center. The locus of the center is given by $2ax + 2by - (a^2 + b^2 + 4) = 0$.

Final Answer

The final answer is \boxed{2ax + 2by - (a^2 + b^2 + 4) = 0}, which corresponds to option (A).