Question

From a point \( (1,0) \) on the circle \( x^2 + y^2 - 2x + 2y + 1 = 0 \), if chords are drawn to this circle, then locus of the poles of these chords with respect to the circle \( x^2 + y^2 = 4 \) is:

• A. \( x = 4 \)
• B. \( x + 2y = 5 \)
• C. \( x^2 + y^2 - x - y = 0 \)
• D. \( 2y^2 = (x + 1) \)

Hint:

For locus of poles, use the pole-chord relations and apply transformation techniques.

Answer 1

The Correct Option is A

To solve this problem, we need to find the locus of the poles of the chords drawn from the point \( (1,0) \) to the given circle equation \( x^2 + y^2 - 2x + 2y + 1 = 0 \). We then analyze these chords with respect to the circle \( x^2 + y^2 = 4 \).
Let's break this down step by step:
First, rewrite the given circle equation:
\( x^2 + y^2 - 2x + 2y + 1 = 0 \).
Completing the square, we have:
\( (x-1)^2 + (y+1)^2 = 1 \).
This represents a circle centered at \( (1,-1) \) with radius 1.

Now, consider the circle \( x^2 + y^2 = 4\). The pole of a chord drawn to a circle is the point from which tangents to the end points of the chord can be drawn. The polar of the point \( (x_1, y_1) \) with respect to the circle \( x^2 + y^2 = 4 \) is given by:
\( x_1 x + y_1 y = 4 \).
Here, we have the given point \( (1,0) \), for which the polar is:
\( x = 4 \).
Since the locus of the poles of all chords from a point to a circle is equivalent to the polar of that point, the locus of the poles with respect to \( x^2 + y^2 = 4 \) is the line:
\( x = 4 \).
Thus, the locus of these poles is the vertical line \( x = 4 \).

Answer 2

Step 1: Identify the Given Circles We are given: 
- Circle 1: \( x^2 + y^2 - 2x + 2y + 1 = 0 \) Completing the square for this circle: \[ x^2 - 2x + y^2 + 2y + 1 = 0 \] \[ (x - 1)^2 + (y + 1)^2 = 1 \] This is a circle with center \( (1, -1) \) and radius 1. 
- Circle 2: \( x^2 + y^2 = 4 \) This is a circle with center \( (0, 0) \) and radius 2. 

Step 2: Understanding the Concept of Pole and Polar If a point \( (x_1, y_1) \) lies on the first circle, the equation of the polar with respect to the second circle is given by: \[ x_1 x + y_1 y = r^2 \] For circle 2 (with radius 2), the polar equation becomes: \[ x_1 x + y_1 y = 4 \] The locus of the poles is this equation rearranged in terms of \(x\) and \(y\). 

Step 3: Finding the Required Locus From the given point \( (1, 0) \), the equation of the polar with respect to the second circle is: \[ 1 \cdot x + 0 \cdot y = 4 \] \[ x = 4 \] 

Step 4: Final Answer 

\[Correct Answer: (1) \ x = 4\]