The co-ordinates of the middle point of the chord cut off on 2x - 5y +18= 0 by the circle `x^2 + y^2 - 6x + 2y - 54 = 0` are (A) (1,1) (B) (2, 4) (C) (4,1) (D) (1, 4)

Options

A. `(1,4)`
B. `(2,4)`
C. `(4,1)`
D. `(1,1)`

Explanation

To find the coordinates of the midpoint of the chord cut off by the line \(2x - 5y + 18 = 0\) on the circle given by the equation \(x^2 + y^2 - 6x + 2y - 54 = 0\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation of the circle in standard form by completing the square.

The given equation is:
\[ x^2 + y^2 - 6x + 2y - 54 = 0 \]

Rearranging gives:
\[ x^2 - 6x + y^2 + 2y = 54 \]

Completing the square for \(x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]

Completing the square for \(y\):
\[ y^2 + 2y = (y + 1)^2 - 1 \]

Substituting back into the equation:
\[ (x - 3)^2 - 9 + (y + 1)^2 - 1 = 54 \]
\[ (x - 3)^2 + (y + 1)^2 = 64 \]

Thus, the center of the circle is \((3, -1)\) and the radius is \(8\).

### Step 2: Find the Midpoint of the Chord
Let the midpoint of the chord be \(M(h, k)\). Since \(M\) lies on the line \(2x - 5y + 18 = 0\), we can substitute \(h\) and \(k\) into this equation:
\[ 2h - 5k + 18 = 0 \]
Rearranging gives:
\[ 5k = 2h + 18 \quad \Rightarrow \quad k = \frac{2h + 18}{5} \]

### Step 3: Find the Slope of the Line and the Perpendicular Slope
The slope of the line \(2x - 5y + 18 = 0\) can be found by rewriting it in slope-intercept form:
\[ 5y = 2x + 18 \quad \Rightarrow \quad y = \frac{2}{5}x + \frac{18}{5} \]
Thus, the slope \(m_1\) is \(\frac{2}{5}\).

The slope of the line connecting the center of the circle \(C(3, -1)\) to the midpoint \(M(h, k)\) is given by:
\[ m_2 = \frac{k - (-1)}{h - 3} = \frac{k + 1}{h - 3} \]

Since the lines are perpendicular, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting \(m_1\) and \(m_2\):
\[ \frac{2}{5} \cdot \frac{k + 1}{h - 3} = -1 \]

### Step 4: Substitute \(k\) and Solve for \(h\)
Substituting \(k = \frac{2h + 18}{5}\) into the perpendicular slope equation:
\[ \frac{2}{5} \cdot \frac{\frac{2h + 18}{5} + 1}{h - 3} = -1 \]

This simplifies to:
\[ \frac{2}{5} \cdot \frac{\frac{2h + 18 + 5}{5}}{h - 3} = -1 \]
\[ \frac{2(2h + 23)}{25(h - 3)} = -1 \]
Cross-multiplying gives:
\[ 2(2h + 23) = -25(h - 3) \]
Expanding both sides:
\[ 4h + 46 = -25h + 75 \]
Combining like terms:
\[ 29h = 29 \quad \Rightarrow \quad h = 1 \]

### Step 5: Find \(k\)
Substituting \(h = 1\) back into the equation for \(k\):
\[ k = \frac{2(1) + 18}{5} = \frac{20}{5} = 4 \]

### Conclusion
The coordinates of the midpoint \(M\) are \((1, 4)\).

Thus, the answer is:
\[ \text{(D) } (1, 4) \]