The polars of two points A(1,3), B(2,-1) w.r.t to circle `x^(2)+y^(2)=9` intersect at C then polar of C w.r.t to the circle is

Options

A. `4x+y-7=0`
B. `4x+y+7=0`
C. `x+y-7=0`
D. `4x-y+7=0`

Answer

To solve the problem, we need to find the polar of the point C, which is the intersection of the polars of points A(1,3) and B(2,-1) with respect to the circle \(x^2 + y^2 = 9\).

### Step-by-Step Solution:

1. **Find the polar of point A(1,3)**:
The equation of the polar of a point \((x_1, y_1)\) with respect to the circle \(x^2 + y^2 = r^2\) is given by:
\[ x_1x + y_1y = r^2 \]
For point A(1,3) and the circle \(x^2 + y^2 = 9\) (where \(r^2 = 9\)):
\[ 1 \cdot x + 3 \cdot y = 9 \]
This simplifies to:
\[ x + 3y = 9 \quad \text{(Equation 1)} \]

2. **Find the polar of point B(2,-1)**:
Similarly, for point B(2,-1):
\[ 2x - 1y = 9 \]
This simplifies to:
\[ 2x - y = 9 \quad \text{(Equation 2)} \]

3. **Solve the system of equations (Equation 1 and Equation 2)**:
We now have the two equations:
\[ x + 3y = 9 \quad \text{(1)} \]
\[ 2x - y = 9 \quad \text{(2)} \]
We can solve these equations simultaneously. First, we can express \(y\) from Equation 1:
\[ y = \frac{9 - x}{3} \]
Substitute \(y\) in Equation 2:
\[ 2x - \left(\frac{9 - x}{3}\right) = 9 \]
Multiply through by 3 to eliminate the fraction:
\[ 6x - (9 - x) = 27 \]
Simplifying gives:
\[ 6x - 9 + x = 27 \]
\[ 7x = 36 \]
\[ x = \frac{36}{7} \]

4. **Find the value of y**:
Substitute \(x = \frac{36}{7}\) back into the expression for \(y\):
\[ y = \frac{9 - \frac{36}{7}}{3} \]
\[ y = \frac{\frac{63 - 36}{7}}{3} = \frac{\frac{27}{7}}{3} = \frac{27}{21} = \frac{9}{7} \]

5. **Coordinates of point C**:
The coordinates of point C are:
\[ C\left(\frac{36}{7}, \frac{9}{7}\right) \]

6. **Find the polar of point C**:
Now, we need to find the polar of point C with respect to the circle. Using the polar formula:
\[ \frac{36}{7}x + \frac{9}{7}y = 9 \]
Multiply through by 7 to eliminate the fraction:
\[ 36x + 9y = 63 \]
This simplifies to:
\[ 4x + y = 7 \]

### Final Answer:
The polar of point C with respect to the circle \(x^2 + y^2 = 9\) is:
\[ 4x + y = 7 \]