Let P(x, y) be a variable point such that `|sqrt((x-1)^(2)+(y-2)^(2))-sqrt((x-5)^(2)+(y-5)^(2))=4` which represents a hyperbola.
Q. If origin is shifted to point `(3, (7)/(2))` and axes are rotated in anticlockwise through an angle `theta`, so that the equation of hyperbola reduces to its standard form `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`, then `theta` equals

Options

A. `tan^(-1)((4)/(3))`
B. `tan^(-1)((3)/(4))`
C. `tan^(-1)((5)/(4))`
D. `tan^(-1)((4)/(5))`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript and derive the necessary equations and values.

### Step 1: Understanding the Given Equation
The equation given is:
\[ |\sqrt{(x-1)^2 + (y-2)^2} - \sqrt{(x-5)^2 + (y-5)^2}| = 4 \]
This represents a hyperbola where the points \(A(1, 2)\) and \(B(5, 5)\) are the foci.

### Step 2: Finding the Center of the Hyperbola
The center \(C\) of the hyperbola is the midpoint of the segment joining the foci \(A\) and \(B\):
\[ C = \left(\frac{1 + 5}{2}, \frac{2 + 5}{2}\right) = \left(3, \frac{7}{2}\right) \]

### Step 3: Finding the Distance Between the Foci
The distance \(d\) between the foci \(A\) and \(B\) can be calculated using the distance formula:
\[ d = \sqrt{(5 - 1)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 4: Relating the Distances to Hyperbola Parameters
For a hyperbola, the distance between the foci is given by \(2c\), where \(c\) is the distance from the center to each focus. Thus:
\[ 2c = 5 \implies c = \frac{5}{2} \]
The given equation states that the difference of distances from any point on the hyperbola to the foci is \(2a\):
\[ 2a = 4 \implies a = 2 \]

### Step 5: Finding the Value of \(b\)
Using the relationship \(c^2 = a^2 + b^2\):
\[ c^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}, \quad a^2 = 2^2 = 4 \]
Thus,
\[ \frac{25}{4} = 4 + b^2 \implies b^2 = \frac{25}{4} - 4 = \frac{25}{4} - \frac{16}{4} = \frac{9}{4} \implies b = \frac{3}{2} \]

### Step 6: Writing the Standard Form of the Hyperbola
The standard form of the hyperbola centered at \(C(3, \frac{7}{2})\) is:
\[ \frac{(x - 3)^2}{2^2} - \frac{\left(y - \frac{7}{2}\right)^2}{\left(\frac{3}{2}\right)^2} = 1 \]
This simplifies to:
\[ \frac{(x - 3)^2}{4} - \frac{\left(y - \frac{7}{2}\right)^2}{\frac{9}{4}} = 1 \]

### Step 7: Finding the Angle of Rotation
The slope \(m\) of the line connecting the foci \(A(1, 2)\) and \(B(5, 5)\) is:
\[ m = \frac{5 - 2}{5 - 1} = \frac{3}{4} \]
The angle \(\theta\) that this line makes with the x-axis is given by:
\[ \tan \theta = m = \frac{3}{4} \]
Thus, the angle \(\theta\) can be found using:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

### Final Answer
Therefore, the value of \(\theta\) is:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]