The polar of (2,3) w.r.t the circle `x^(2)+y^(2)-4x-6y+2=0` is a) a tangent b) a diameter c) a chord of contact d) not existing

Options

A. a tangent
B. a diameter
C. a chord of contact
D. not existing

Answer

To find the polar of the point (2, 3) with respect to the circle given by the equation \(x^2 + y^2 - 4x - 6y + 2 = 0\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
The given circle equation is:
\[ x^2 + y^2 - 4x - 6y + 2 = 0 \]
We can rewrite this in standard form by completing the square.

### Step 2: Completing the Square
1. For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
2. For \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

Now substituting these back into the equation:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 + 2 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 3)^2 - 11 = 0 \]
Thus, the standard form of the circle is:
\[ (x - 2)^2 + (y - 3)^2 = 11 \]
This indicates that the center of the circle is at (2, 3) and the radius is \(\sqrt{11}\).

### Step 3: Identify the Point and Circle Relationship
The point (2, 3) is the center of the circle. The polar of a point with respect to a circle is defined only if the point is outside the circle. Since (2, 3) is the center of the circle, it lies inside the circle.

### Step 4: Determine the Polar
According to the properties of polar lines:
- If the point lies inside the circle, the polar does not exist.

### Conclusion
Since the point (2, 3) is the center of the circle, the polar of this point with respect to the circle does not exist.

Thus, the answer is:
**d) not existing**

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