Question

The lines \( 2x - 3y - 5 = 0 \) and \( 3x - 4y - 7 = 0 \) are diameters of a circle of area 154 square units, then the equation of the circle is:

• A. \( x^2 + y^2 + 2x - 2y - 62 = 0 \)
• B. \( x^2 + y^2 + 2x - 2y - 47 = 0 \)
• C. \( x^2 + y^2 - 2x + 2y - 47 = 0 \)
• D. \( x^2 + y^2 - 2x + 2y - 62 = 0 \)

Hint:

To find the equation of a circle given two diameters, first find the midpoint of the diameters (the center) and the radius (half the length of the diameters).

Answer 1

The Correct Option is C

Step 1: Understanding the equation of a circle.
The general equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2, \] where \( (h, k) \) is the center and \( r \) is the radius. The center of the circle is the midpoint of the two diameters, and the radius is half the length of the diameters.
Step 2: Finding the midpoint of the diameters.
The equations of the diameters are given by: \[ 2x - 3y - 5 = 0 \quad \text{(1)} \] and \[ 3x - 4y - 7 = 0 \quad \text{(2)}. \] To find the midpoint, we need to solve these two linear equations simultaneously. Solving equation (1) and (2) by substitution or elimination gives us the center of the circle.
Step 3: Finding the radius.
The area of the circle is given as 154 square units, and the area of a circle is: \[ \text{Area} = \pi r^2. \] Thus, \[ \pi r^2 = 154. \] Solving for \( r \), we get: \[ r^2 = \frac{154}{\pi} \approx 49. \] Thus, \( r = 7 \).
Step 4: Writing the equation of the circle.
Now that we have the center and the radius, we can substitute these values into the equation of the circle. After solving for the center and substituting into the general equation, we obtain the equation of the circle: \[ x^2 + y^2 - 2x + 2y - 47 = 0. \]
Step 5: Conclusion.
Therefore, the equation of the circle is \( x^2 + y^2 - 2x + 2y - 47 = 0 \), and the correct answer is (c).