Key Concepts and Formulas

• The general equation of a circle is given by $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center of the circle is $(-g, -f)$.
• The midpoint formula states that the midpoint $M(x_m, y_m)$ of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.
• A diameter of a circle is a line segment that passes through the center of the circle and has its endpoints on the circle. The center of the circle is the midpoint of any diameter.

Step-by-Step Solution

Step 1: Identify the given information and the goal.

We are given the equation of a circle $x^2 + y^2 + 2x + 4y - 3 = 0$, and a point $P(1, 0)$ on the circle. We want to find the coordinates of the point $Q(\alpha, \beta)$ that is diametrically opposite to $P$.

Why? This step clarifies what we know and what we need to find, setting the stage for using the properties of circles and diameters.

Step 2: Find the center of the circle.

The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, and its center is $(-g, -f)$. Comparing this to the given equation $x^2 + y^2 + 2x + 4y - 3 = 0$, we have $2g = 2$ and $2f = 4$. Therefore, $g = 1$ and $f = 2$. The center of the circle, $C$, is thus $(-g, -f) = (-1, -2)$.

Why? The center is the midpoint of the diameter, so we need to find it to use the midpoint formula.

Step 3: Apply the midpoint formula.

Since $C(-1, -2)$ is the midpoint of the diameter with endpoints $P(1, 0)$ and $Q(\alpha, \beta)$, we can use the midpoint formula: $\frac{1 + \alpha}{2} = -1 \quad \text{and} \quad \frac{0 + \beta}{2} = -2$

Why? This step applies the core geometric principle relating the center and endpoints of a diameter, setting up the equations to solve for the unknown coordinates.

Step 4: Solve for the coordinates of point $Q$.

Solving the equations from Step 3: For the x-coordinate: $\frac{1 + \alpha}{2} = -1$ $1 + \alpha = -2$ $\alpha = -3$ For the y-coordinate: $\frac{0 + \beta}{2} = -2$ $\beta = -4$ Therefore, the coordinates of point $Q$ are $(-3, -4)$.

Why? This step performs the algebraic manipulation to isolate the unknowns, leading to the solution for the coordinates of the opposite endpoint.

Common Mistakes & Tips

• Ensure the coefficients of $x^2$ and $y^2$ are both 1 before identifying $g$ and $f$ to find the center.
• Double-check your arithmetic when solving for $\alpha$ and $\beta$ to avoid errors.
• Remember that the center of the circle is the midpoint of the diameter, not just any point on the circle.

Summary

We found the coordinates of the point diametrically opposite to $P(1, 0)$ on the circle $x^2 + y^2 + 2x + 4y - 3 = 0$ by first determining the center of the circle to be $(-1, -2)$, and then using the midpoint formula with the given point and the circle's center to find the coordinates of the opposite point. The coordinates of the diametrically opposite point are $(-3, -4)$.

Final Answer

The final answer is \boxed{(-3, -4)}, which corresponds to option (C).