Key Concepts and Formulas

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

Step-by-Step Solution

Step 1: Determine the Radius of the Circle

• Understanding the Goal: We are given the area of the circle and need to find its radius. This radius will be used in the circle's equation.
• Formula: The area of a circle $A$ is given by $A = \pi r^2$.
• Given Information: The area of the circle is 154 sq. units.
• Calculation: We have the equation: $A = \pi r^2$ Substituting the given area and using $\pi = \frac{22}{7}$: $154 = \frac{22}{7} r^2$ Multiplying both sides by $\frac{7}{22}$ to isolate $r^2$: $r^2 = 154 \times \frac{7}{22}$ $r^2 = 7 \times 7$ $r^2 = 49$ Taking the square root (and considering only the positive root for the radius): $r = \sqrt{49}$ $r = 7$
• Explanation: We used the area of the circle and the formula $A = \pi r^2$ to solve for the radius $r$. Squaring the radius gives $r^2 = 49$, which we will use directly in the equation of the circle.

Step 2: Find the Center of the Circle

• Understanding the Goal: We are given two lines that are diameters of the circle. The intersection of these lines gives us the center of the circle.
• Given Information:
  1. First diameter: $2x - 3y = 5$ (Equation 1)
  2. Second diameter: $3x - 4y = 7$ (Equation 2)
• Method: We will solve the system of linear equations using the elimination method.
• Calculation: Multiply Equation 1 by 3: $3 \times (2x - 3y) = 3 \times 5 \implies 6x - 9y = 15 \quad (\text{Equation 3})$ Multiply Equation 2 by 2: $2 \times (3x - 4y) = 2 \times 7 \implies 6x - 8y = 14 \quad (\text{Equation 4})$ Subtract Equation 4 from Equation 3 to eliminate $x$: $(6x - 9y) - (6x - 8y) = 15 - 14$ $6x - 9y - 6x + 8y = 1$ $-y = 1$ $y = -1$ Substitute the value of $y = -1$ back into Equation 1: $2x - 3(-1) = 5$ $2x + 3 = 5$ $2x = 5 - 3$ $2x = 2$ $x = 1$
• Result: The intersection point is $(1, -1)$. Therefore, the center of the circle $(h, k)$ is $(1, -1)$.
• Explanation: Since the diameters intersect at the center of the circle, we solve the system of equations formed by the two diameters to find the coordinates of the center.

Step 3: Formulate the Equation of the Circle

• Understanding the Goal: With the center $(h,k)$ and radius $r$ determined, we can now write the equation of the circle.
• Formula: The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$.
• Known Values:
  • Center $(h, k) = (1, -1)$
  • Radius squared $r^2 = 49$ (from Step 1)
• Calculation: Substitute these values into the standard equation: $(x - 1)^2 + (y - (-1))^2 = 49$ $(x - 1)^2 + (y + 1)^2 = 49$
• Explanation: We plug the center coordinates and the square of the radius into the standard circle equation.

Step 4: Simplify the Equation and Match with Options

• Understanding the Goal: The options are given in the general form of the circle's equation. We need to expand and rearrange our equation to match this format.
• Calculation: Expand the squared terms: $(x - 1)^2 = x^2 - 2x + 1$ $(y + 1)^2 = y^2 + 2y + 1$ Substitute these expansions back into the equation from Step 3: $(x^2 - 2x + 1) + (y^2 + 2y + 1) = 49$ Combine the constant terms: $x^2 + y^2 - 2x + 2y + 2 = 49$ Move the constant term to the right side of the equation: $x^2 + y^2 - 2x + 2y = 49 - 2$ $x^2 + y^2 - 2x + 2y = 47$
• Matching with Options: Comparing this result with the given options: (A) ${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,47$ (B) ${x^2}\, + \,{y^2} + \,2x\, - \,2y\,\, = \,62$ (C) ${x^2}\, + \,{y^2} + \,2x\, - \,2y\,\, = \,47$ (D) ${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,62$ Our derived equation matches option (A).

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when expanding the squared terms and substituting values. A common mistake is to incorrectly expand $(y+1)^2$ or $(x-1)^2$.
• Solving Linear Equations: Double-check your work when solving the system of linear equations to find the center. A small error here will propagate through the rest of the solution.
• Using the correct formula: Ensure you remember the correct formulas for the area and the equation of a circle.

Summary

We first calculated the radius of the circle using the given area. Then, we found the center of the circle by solving the system of equations formed by the two diameters. Finally, we substituted the center coordinates and the squared radius into the standard equation of a circle and simplified to match one of the given options.

The final answer is \boxed{47}, which corresponds to option (A).