Find the polar of the point (1,2) w.r.t the circle `x^2+y^2=7` .

Answer

To find the polar of the point (1, 2) with respect to the circle given by the equation \(x^2 + y^2 = 7\), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle and Point**:
The equation of the circle is \(x^2 + y^2 = 7\). The point given is \(P(1, 2)\).

2. **Determine \(R^2\)**:
From the circle's equation, we can see that \(R^2 = 7\).

3. **Use the Polar Equation**:
The polar of a point \((x_1, y_1)\) with respect to the circle \(x^2 + y^2 = R^2\) is given by the equation:
\[ xx_1 + yy_1 = R^2 \]
Here, \(x_1 = 1\) and \(y_1 = 2\).

4. **Substitute the Values**:
Substitute \(x_1\), \(y_1\), and \(R^2\) into the polar equation:
\[ x(1) + y(2) = 7 \]
This simplifies to:
\[ x + 2y = 7 \]

5. **Final Polar Equation**:
The polar of the point (1, 2) with respect to the circle \(x^2 + y^2 = 7\) is:
\[ x + 2y = 7 \]