A rod AB of length 4 units moves horizontally with its left end A always on the circle `x^(2)+y^(2)-4x-18y-29=0` then the locus of the other end B is

Options

A. `x^(2)+y^(2)-12x-8y+3=0`
B. `x^(2)+y^(2)-12x-18y+3=0`
C. `x^(2)+y^(2)+4x-7y-29=0`
D. `x^(2)+y^(2)-4x-16y+19=0`

Answer

To solve the problem, we need to find the locus of the other end B of a rod AB of length 4 units, where the left end A is always on the given circle.

### Step 1: Identify the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 18y - 29 = 0 \]

### Step 2: Rewrite the equation in standard form
To rewrite the equation in standard form, we complete the square for both x and y.

1. For \(x\):
\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \]

2. For \(y\):
\[ y^2 - 18y \rightarrow (y - 9)^2 - 81 \]

Substituting these into the equation gives:
\[ (x - 2)^2 - 4 + (y - 9)^2 - 81 - 29 = 0 \]
\[ (x - 2)^2 + (y - 9)^2 - 114 = 0 \]
\[ (x - 2)^2 + (y - 9)^2 = 114 \]

### Step 3: Identify the center and radius of the circle
From the standard form, we can identify:
- Center: \(C(2, 9)\)
- Radius: \(r = \sqrt{114}\)

### Step 4: Determine the position of point A
Let point A be at coordinates \((x_A, y_A)\) on the circle. Since A is on the circle, it satisfies:
\[ (x_A - 2)^2 + (y_A - 9)^2 = 114 \]

### Step 5: Determine the position of point B
Since the length of the rod AB is 4 units and it moves horizontally, the coordinates of point B can be expressed as:
\[ B(x_B, y_B) = (x_A + 4, y_A) \]

### Step 6: Substitute to find the locus of point B
Substituting \(x_B = x_A + 4\) into the equation of the circle:
\[ ((x_B - 4) - 2)^2 + (y_A - 9)^2 = 114 \]
This simplifies to:
\[ (x_B - 6)^2 + (y_A - 9)^2 = 114 \]

### Step 7: Write the equation for the locus of B
The locus of point B is another circle with:
- Center: \(C'(6, 9)\)
- Radius: \(\sqrt{114}\)

Thus, the equation of the locus of point B is:
\[ (x - 6)^2 + (y - 9)^2 = 114 \]

### Final Answer
The locus of the other end B is given by the equation:
\[ (x - 6)^2 + (y - 9)^2 = 114 \]