If `x+iy=sqrt(phi+iy)`, where `i=sqrt(-1)" and "phi` and `Psi` are non-zero real parameters, then `phi" and "Psi` are constants, represents two system of rectangular hyperbola which intersect at an angle of :

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(2)`

Answer

To solve the problem, we start with the equation given in the question:

\[ x + iy = \sqrt{\phi + i\Psi} \]

where \( i = \sqrt{-1} \) and \( \phi \) and \( \Psi \) are non-zero real parameters.

### Step 1: Square both sides

To eliminate the square root, we square both sides of the equation:

\[ (x + iy)^2 = \phi + i\Psi \]

### Step 2: Expand the left-hand side

Expanding the left-hand side using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):

\[ x^2 + 2xyi - y^2 = \phi + i\Psi \]

### Step 3: Separate real and imaginary parts

Now, we separate the real and imaginary parts:

1. Real part: \( x^2 - y^2 = \phi \)
2. Imaginary part: \( 2xy = \Psi \)

### Step 4: Identify the equations of hyperbolas

From the equations obtained, we can rewrite them as:

1. \( x^2 - y^2 = \phi \) (Equation of a hyperbola)
2. \( 2xy = \Psi \) (Equation of another hyperbola)

These represent two rectangular hyperbolas.

### Step 5: Find the angle of intersection

The angle \( \theta \) between two curves given by their equations can be found using the formula:

\[ \tan(\theta) = \left| \frac{f'(x_0)}{g'(x_0)} \right| \]

where \( f(x) \) and \( g(x) \) are the equations of the curves, and \( (x_0, y_0) \) is the point of intersection.

For the hyperbolas:

1. From \( x^2 - y^2 = \phi \), differentiating gives:
\[ 2x - 2yy' = 0 \implies y' = \frac{x}{y} \]

2. From \( 2xy = \Psi \), differentiating gives:
\[ 2y + 2xy' = 0 \implies y' = -\frac{y}{x} \]

### Step 6: Calculate the angle of intersection

Now, substituting \( y' \) from both equations into the tangent formula:

\[ \tan(\theta) = \left| \frac{\frac{x}{y}}{-\frac{y}{x}} \right| = \left| -\frac{x^2}{y^2} \right| \]

The angle \( \theta \) between the two hyperbolas can be determined using the formula for the angle between two curves:

\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \]

where \( m_1 \) and \( m_2 \) are the slopes of the tangents at the point of intersection.

### Step 7: Conclusion

The two hyperbolas intersect at right angles since their slopes are negative reciprocals of each other. Therefore, the angle of intersection is:

\[ \theta = 90^\circ \]

### Final Answer

The two systems of rectangular hyperbolas intersect at an angle of \( 90^\circ \).

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