Question

The co-ordinates of the mid-point of the chord cut off by the line \( 2x - 5y + 18 = 0 \) by the circle \( x^2 + y^2 - 6x + 2y - 54 = 0 \) are

• A. \( (1, 4) \)
• B. \( (2, 4) \)
• C. \( (4, 1) \)
• D. \( (1, 1) \)

Hint:

To find the mid-point of a chord, use the formula for the coordinates of the mid-point given the center and equation of the circle.

Answer 1

The Correct Option is A

Step 1: Write the equation of the line and circle.
We are given the line \( 2x - 5y + 18 = 0 \) and the equation of the circle \( x^2 + y^2 - 6x + 2y - 54 = 0 \).
Step 2: Find the mid-point of the chord.
To find the mid-point of the chord cut off by the line and the circle, we use the formula for the mid-point of a chord: \[ \text{Mid-point} = \left( \frac{-h}{a}, \frac{-k}{b} \right) \] where \( h = -2 \) and \( k = 6 \) (the values of the circle's center).
Step 3: Conclusion.
The correct co-ordinates of the mid-point are \( (1, 4) \).