A and B are points in the plane such that `(PA)/ (PB)` =k (constant) for all points P on a circle, then the value of k cannot be equal to ..............

Answer

To solve the problem, we need to analyze the condition given: \((PA)/(PB) = k\) for all points \(P\) on a circle. Here are the step-by-step details:

### Step 1: Understanding the Relationship
We are given two fixed points \(A\) and \(B\) in the plane, and we need to find the values of \(k\) for which the ratio of distances from any point \(P\) on a circle to points \(A\) and \(B\) remains constant.

### Step 2: Define Points
Let’s denote the coordinates of points \(A\) and \(B\) as:
- \(A = (\alpha_1, \beta_1)\)
- \(B = (\alpha_2, \beta_2)\)

Let \(P\) be any point on the circle with coordinates \(P = (x, y)\).

### Step 3: Write the Distance Formulas
The distances from point \(P\) to points \(A\) and \(B\) are given by:
- \(PA = \sqrt{(x - \alpha_1)^2 + (y - \beta_1)^2}\)
- \(PB = \sqrt{(x - \alpha_2)^2 + (y - \beta_2)^2}\)

### Step 4: Set Up the Ratio
We set up the equation based on the given ratio:
\[ \frac{PA}{PB} = k \]
Squaring both sides gives:
\[ \frac{(PA)^2}{(PB)^2} = k^2 \]
This leads to:
\[ \frac{(x - \alpha_1)^2 + (y - \beta_1)^2}{(x - \alpha_2)^2 + (y - \beta_2)^2} = k^2 \]

### Step 5: Cross-Multiply
Cross-multiplying gives:
\[ (x - \alpha_1)^2 + (y - \beta_1)^2 = k^2 \left((x - \alpha_2)^2 + (y - \beta_2)^2\right) \]

### Step 6: Expand Both Sides
Expanding both sides results in:
\[ (x^2 - 2\alpha_1 x + \alpha_1^2 + y^2 - 2\beta_1 y + \beta_1^2) = k^2 (x^2 - 2\alpha_2 x + \alpha_2^2 + y^2 - 2\beta_2 y + \beta_2^2) \]

### Step 7: Rearranging Terms
Rearranging gives:
\[ (1 - k^2)(x^2 + y^2) + (-2\alpha_1 + 2k^2\alpha_2)x + (-2\beta_1 + 2k^2\beta_2)y + (\alpha_1^2 + \beta_1^2 - k^2(\alpha_2^2 + \beta_2^2)) = 0 \]

### Step 8: Identifying the Circle Condition
For this equation to represent a circle, the coefficient of \(x^2 + y^2\) must be non-zero. Thus:
\[ 1 - k^2 \neq 0 \]
This implies:
\[ k^2 \neq 1 \]
Therefore, \(k\) cannot be equal to \(1\) or \(-1\).

### Conclusion
The value of \(k\) cannot be equal to \(1\) or \(-1\).

### Final Answer
The value of \(k\) cannot be equal to \( \pm 1 \).

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