Circles

Step 1: Find the pole of the given line with respect to the circle .
The equation of the circle is .
Comparing with , we have:



Let the pole of the line with respect to the circle be .
The polar of is .
Substituting the values of :

Rearrange terms to match the form of a linear equation:
.
This polar equation must represent the given line .
Comparing the coefficients of , , and the constant term:
From the first two parts of the equality, let be the common ratio:

Substitute these expressions for and into the third part of the equality:
Move terms involving to one side and constants to the other: Now find the coordinates of the pole : So, the pole is .
Step 2: Use the condition that the pole lies on the line .
Substitute the coordinates of the pole into the equation of the line : Multiply the entire equation by to clear the denominator (assuming ): Combine the constant terms and the terms involving : Step 3: Find the polar of the point with respect to the circle .
We found that . So, the point is .
The circle is , with .
The polar of a point with respect to is .
Substitute : Combine like terms: Multiplying by -1, we get: This matches Option (1).

Step 1: Understand the condition for conjugate points.
Two points and are said to be conjugate with respect to a circle if the polar of one point passes through the other point. The condition for conjugacy is .
Step 2: Identify the circle equation and the given points.
The given circle equation is , which can be written as . Comparing this to the general form , we have:



The two given points are and .
Step 3: Apply the conjugate points condition.
Substitute the coordinates of the points and the coefficients of the circle into the conjugacy condition:
Rearrange the terms to find the relationship between and : Step 4: Calculate the required expression .
We need to find the value of .
Notice that is simply times the expression : Substitute the value of from Step 3: