The midpoint of chord formed by the polar of (-9,12) w.r.t `x^(2)+y^(2)=100` is

Options

A. `(4,-4/3)`
B. `(-4,16/3)`
C. `(-4,16/9)`
D. `(4,16/3)`

Answer

To find the midpoint of the chord formed by the polar of the point (-9, 12) with respect to the circle given by the equation \(x^2 + y^2 = 100\), we can follow these steps:

### Step 1: Identify the given information
The equation of the circle is:
\[ x^2 + y^2 = 100 \]
The point given is:
\[ (x_1, y_1) = (-9, 12) \]

### Step 2: Write the equation of the polar line
The equation of the polar line of a point \((x_1, y_1)\) with respect to the circle \(x^2 + y^2 = r^2\) is given by:
\[ x x_1 + y y_1 - r^2 = 0 \]
Here, \(r^2 = 100\). Therefore, substituting the values:
\[ x(-9) + y(12) - 100 = 0 \]
This simplifies to:
\[ -9x + 12y - 100 = 0 \]
or
\[ 9x - 12y + 100 = 0 \]

### Step 3: Identify coefficients for the midpoint formula
From the equation \(Lx + My + N = 0\), we can identify:
- \(L = -9\)
- \(M = 12\)
- \(N = -100\)

### Step 4: Use the midpoint formula
The coordinates of the midpoint \((x_m, y_m)\) of the chord can be found using the formulas:
\[ x_m = \frac{-LN}{L^2 + M^2} \]
\[ y_m = \frac{-MN}{L^2 + M^2} \]

### Step 5: Calculate \(L^2 + M^2\)
First, calculate \(L^2 + M^2\):
\[ L^2 = (-9)^2 = 81 \]
\[ M^2 = 12^2 = 144 \]
\[ L^2 + M^2 = 81 + 144 = 225 \]

### Step 6: Calculate \(x_m\)
Now substitute \(L\), \(M\), and \(N\) into the formula for \(x_m\):
\[ x_m = \frac{-(-9)(-100)}{225} = \frac{900}{225} = 4 \]

### Step 7: Calculate \(y_m\)
Now substitute into the formula for \(y_m\):
\[ y_m = \frac{-12(-100)}{225} = \frac{1200}{225} = \frac{1200 \div 75}{225 \div 75} = \frac{16}{3} \]

### Step 8: Final result
Thus, the midpoint of the chord is:
\[ \left(4, \frac{16}{3}\right) \]

### Summary
The midpoint of the chord formed by the polar of the point (-9, 12) with respect to the circle \(x^2 + y^2 = 100\) is:
\[ \boxed{\left(4, \frac{16}{3}\right)} \]