Key Concepts and Formulas

• Equation of a Circle: The equation of a circle with center $(h, k)$ and radius $r$ is given by $(x - h)^2 + (y - k)^2 = r^2$.
• Intersection of Diameters: The intersection point of any two diameters of a circle is the center of the circle.
• Area of a Circle: The area $A$ of a circle with radius $r$ is given by $A = \pi r^2$.

Step-by-Step Solution

Step 1: Find the Center of the Circle

Why this step? The center of the circle is the intersection point of its diameters. We are given the equations of two diameters, so we need to solve the system of equations to find their intersection point, which will be the center of the circle.

We have the following system of linear equations:

1. $3x - 4y - 7 = 0$
2. $2x - 3y - 5 = 0$

We can rewrite these equations as:

1. $3x - 4y = 7$
2. $2x - 3y = 5$

Multiply the first equation by 2 and the second equation by 3 to eliminate $x$:

1. $2(3x - 4y) = 2(7) \Rightarrow 6x - 8y = 14$
2. $3(2x - 3y) = 3(5) \Rightarrow 6x - 9y = 15$

Subtract the first modified equation from the second: $(6x - 9y) - (6x - 8y) = 15 - 14$ $-9y + 8y = 1$ $-y = 1$ $y = -1$

Substitute $y = -1$ into the equation $2x - 3y = 5$: $2x - 3(-1) = 5$ $2x + 3 = 5$ $2x = 2$ $x = 1$

The intersection point is $(1, -1)$. Therefore, the center of the circle is $(h, k) = (1, -1)$.

Step 2: Find the Radius of the Circle

Why this step? We are given the area of the circle, which is related to the radius by the formula $A = \pi r^2$. We can use this to find the radius.

The area of the circle is given as $49\pi$. Using the formula $A = \pi r^2$: $49\pi = \pi r^2$

Divide both sides by $\pi$: $49 = r^2$

Take the square root of both sides (since the radius must be positive): $r = \sqrt{49} = 7$

Therefore, the radius of the circle is $r = 7$.

Step 3: Formulate the Equation of the Circle

Why this step? We now know the center $(h, k) = (1, -1)$ and the radius $r = 7$. We can substitute these values into the standard equation of a circle to find the equation of this particular circle.

The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$. Substituting the values we found: $(x - 1)^2 + (y - (-1))^2 = 7^2$ $(x - 1)^2 + (y + 1)^2 = 49$

Expand the equation: $(x^2 - 2x + 1) + (y^2 + 2y + 1) = 49$ $x^2 - 2x + 1 + y^2 + 2y + 1 = 49$ $x^2 + y^2 - 2x + 2y + 2 = 49$ $x^2 + y^2 - 2x + 2y - 47 = 0$

Step 4: Compare with Options

The equation of the circle is $x^2 + y^2 - 2x + 2y - 47 = 0$. This matches option (A).

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs when substituting the center coordinates into the circle equation. For example, $(y - (-1))^2 = (y + 1)^2$.
• Using r vs. r^2: Remember to use $r^2$ in the equation of the circle, not just $r$.
• Algebraic Errors: Carefully expand the binomials $(x-1)^2$ and $(y+1)^2$ to avoid errors.

Summary

We found the center of the circle by solving the system of equations formed by the two diameters. Then, we determined the radius using the area of the circle. Finally, we plugged the center and radius into the standard equation of a circle to obtain the circle's equation: $x^2 + y^2 - 2x + 2y - 47 = 0$, which corresponds to option (A).

The final answer is $\boxed{A}$.