The angle of intersection between the curves `y^2=4x" and "x^2=32y` at point (16,8) is

Options

A. `tan^(-1)(3/5)`
B. `tan^(-1)(4/5)`
C. `pi `
D. `(pi)/(2)`

Answer

To find the angle of intersection between the curves \( y^2 = 4x \) and \( x^2 = 32y \) at the point \( (16, 8) \), we will follow these steps:

### Step 1: Verify the point of intersection
First, we need to confirm that the point \( (16, 8) \) lies on both curves.

1. For the curve \( y^2 = 4x \):
\[ 8^2 = 4 \times 16 \implies 64 = 64 \quad \text{(True)} \]

2. For the curve \( x^2 = 32y \):
\[ 16^2 = 32 \times 8 \implies 256 = 256 \quad \text{(True)} \]

Since both equations are satisfied, \( (16, 8) \) is indeed the point of intersection.

### Step 2: Find the slope of the tangent to the first curve
For the curve \( y^2 = 4x \), we differentiate implicitly to find the slope \( m_1 \).

1. Differentiate:
\[ 2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]

2. Substitute \( y = 8 \):
\[ m_1 = \frac{2}{8} = \frac{1}{4} \]

### Step 3: Find the slope of the tangent to the second curve
For the curve \( x^2 = 32y \), we differentiate implicitly to find the slope \( m_2 \).

1. Differentiate:
\[ 2x = 32 \frac{dy}{dx} \implies \frac{dy}{dx} = \frac{2x}{32} = \frac{x}{16} \]

2. Substitute \( x = 16 \):
\[ m_2 = \frac{16}{16} = 1 \]

### Step 4: Calculate the angle of intersection
The angle \( \theta \) between the two curves can be found using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Substituting the values of \( m_1 \) and \( m_2 \):
\[ \tan \theta = \left| \frac{\frac{1}{4} - 1}{1 + \frac{1}{4} \cdot 1} \right| = \left| \frac{\frac{1}{4} - \frac{4}{4}}{1 + \frac{1}{4}} \right| = \left| \frac{-\frac{3}{4}}{\frac{5}{4}} \right| = \frac{3}{5} \]

### Step 5: Find the angle \( \theta \)
To find \( \theta \):
\[ \theta = \tan^{-1} \left( \frac{3}{5} \right) \]

### Conclusion
The angle of intersection between the curves \( y^2 = 4x \) and \( x^2 = 32y \) at the point \( (16, 8) \) is \( \tan^{-1} \left( \frac{3}{5} \right) \).

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