From the point P(2, 3) tangents PA,PB are drawn to the circle `x^2+y^2-6x+8y-1=0`. The equation to the line joining the mid points of PA and PB is

Options

A. `x-7y+7=0`
B. `x+7y-5=0`
C. `x+7y+5=0`
D. `x-7y-6=0 `

Answer

To solve the problem step by step, we will find the equation of the line joining the midpoints of the tangents PA and PB drawn from the point P(2, 3) to the circle given by the equation \( x^2 + y^2 - 6x + 8y - 1 = 0 \).

### Step 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 - 6x + 8y - 1 = 0 \]
We can rewrite it in standard form by completing the square.

### Step 2: Completing the Square
For \(x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
For \(y\):
\[ y^2 + 8y = (y + 4)^2 - 16 \]
Substituting these back into the equation:
\[ (x - 3)^2 - 9 + (y + 4)^2 - 16 - 1 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y + 4)^2 = 26 \]
Thus, the center of the circle \(C\) is at \((3, -4)\) and the radius \(r\) is \(\sqrt{26}\).

### Step 3: Find the Midpoint of Line Segment CP
The coordinates of point \(P\) are \((2, 3)\) and the center \(C\) is \((3, -4)\). The midpoint \(M\) of segment \(CP\) can be calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 3}{2}, \frac{3 - 4}{2} \right) = \left( \frac{5}{2}, -\frac{1}{2} \right) \]

### Step 4: Find the Slope of Line CP
The slope \(m_{CP}\) of line segment \(CP\) is given by:
\[ m_{CP} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 3}{3 - 2} = \frac{-7}{1} = -7 \]

### Step 5: Find the Slope of the Required Line
The slope of the line joining the midpoints \(M\) and perpendicular to \(CP\) is the negative reciprocal of \(m_{CP}\):
\[ m_{L} = -\frac{1}{m_{CP}} = \frac{1}{7} \]

### Step 6: Use Point-Slope Form to Find the Equation of Line L
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting the coordinates of \(M\) and the slope \(m_L\):
\[ y + \frac{1}{2} = \frac{1}{7} \left( x - \frac{5}{2} \right) \]

### Step 7: Simplify the Equation
Multiply through by 7 to eliminate the fraction:
\[ 7y + \frac{7}{2} = x - \frac{5}{2} \]
Rearranging gives:
\[ 7y - x + 6 = 0 \]
Or:
\[ x - 7y - 6 = 0 \]

### Final Answer
The equation of the line joining the midpoints of PA and PB is:
\[ x - 7y - 6 = 0 \]