An equation chord joining the points `(1,2) and (2,-1)` subtends an angle of `pi//4` at any point on the circumference is

Options

A. `x ^(2) + y ^(2) - 15 =0`
B. `x ^(2) + y ^(2) - 6x - 2y +5 =0`
C. `x ^(2) + y ^(2) + 6x + 2y - 15 =0`
D. `x ^(2) + y ^(2) - 2x - 4y + 4=0`

Answer

To solve the problem, we need to find the equation of the locus of points that subtend an angle of \(\frac{\pi}{4}\) at the chord joining the points \((1, 2)\) and \((2, -1)\).

### Step 1: Identify the Points
The points given are \(A(1, 2)\) and \(B(2, -1)\).

### Step 2: Find the Slopes of the Lines
Let \(P(h, k)\) be any point on the circumference that subtends an angle of \(\frac{\pi}{4}\) at the chord \(AB\).

The slope of line \(PA\) (from point \(P\) to point \(A\)) is given by:
\[ m_1 = \frac{k - 2}{h - 1} \]

The slope of line \(PB\) (from point \(P\) to point \(B\)) is given by:
\[ m_2 = \frac{k + 1}{h - 2} \]

### Step 3: Use the Angle Formula
The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be calculated using the formula:
\[ \tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]
Given that \(\theta = \frac{\pi}{4}\), we have \(\tan\left(\frac{\pi}{4}\right) = 1\). Therefore:
\[ 1 = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]

### Step 4: Substitute the Slopes into the Equation
Substituting \(m_1\) and \(m_2\):
\[ 1 = \left|\frac{\frac{k - 2}{h - 1} - \frac{k + 1}{h - 2}}{1 + \frac{k - 2}{h - 1} \cdot \frac{k + 1}{h - 2}}\right| \]

### Step 5: Simplify the Expression
Cross-multiplying gives:
\[ 1 + m_1 m_2 = |m_1 - m_2| \]
This leads to two cases, but we can simplify it directly to find the locus.

### Step 6: Find the Locus
After simplification, we will arrive at the equation of the locus. The equation will turn out to be of the form:
\[ x^2 + y^2 = 5 \]

### Step 7: Consider the Negative Case
If we consider the negative case from the angle formula, we will derive another equation:
\[ x^2 + y^2 - 6x - 2y + 5 = 0 \]

### Final Result
Thus, the equations of the circles that subtend an angle of \(\frac{\pi}{4}\) at the chord joining the points \((1, 2)\) and \((2, -1)\) are:
1. \(x^2 + y^2 = 5\)
2. \(x^2 + y^2 - 6x - 2y + 5 = 0\)