Question

The two diameters of a circle are segments of the straight lines x - y = 5 and 2x + y = 4. If the radius of the circle is 5, then the equation of the circle is

• A. x²+ y²-6x+4y=12
• B. x²+ y²-3x+2y=12
• C. x²+ y²-6x+2y=12
• D. x²+ y²-8x+6y-18=0
• E. x²+ y²-8x+6y-7=0

Answer 1

The Correct Option is A

Step 1: Find the center of the circle  
The two given diameters are segments of the lines: \[ x - y = 5 \] \[ 2x + y = 4 \] The center of the circle is the intersection of these two lines.

Step 2: Solve for intersection point 
Solving \( x - y = 5 \) for \( x \): \[ x = y + 5 \] Substituting into \( 2x + y = 4 \): \[ 2(y + 5) + y = 4 \] \[ 2y + 10 + y = 4 \] \[ 3y = -6 \] \[ y = -2 \] Substituting \( y = -2 \) into \( x = y + 5 \): \[ x = -2 + 5 = 3 \] So, the center of the circle is \( (3, -2) \).

Step 3: Write the equation of the circle 
The standard equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Given \( (h, k) = (3, -2) \) and radius \( r = 5 \): \[ (x - 3)^2 + (y + 2)^2 = 25 \] Expanding: \[ x^2 - 6x + 9 + y^2 + 4y + 4 = 25 \] \[ x^2 + y^2 - 6x + 4y + 13 = 25 \] \[ x^2 + y^2 - 6x + 4y = 12 \]

Final Answer: The equation of the circle is \( x^2 + y^2 - 6x + 4y = 12 \).