Find the angle at which the parabolas `y^2=4x` and `x^2=32 y` intersect.

Explanation

To find the angle at which the parabolas \( y^2 = 4x \) and \( x^2 = 32y \) intersect, we will follow these steps:

### Step 1: Find the Points of Intersection
We have two equations:
1. \( y^2 = 4x \) (Equation 1)
2. \( x^2 = 32y \) (Equation 2)

From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = \frac{y^2}{4} \]

Substituting this expression for \( x \) into Equation 2 gives:
\[ \left(\frac{y^2}{4}\right)^2 = 32y \]
\[ \frac{y^4}{16} = 32y \]
Multiplying both sides by 16 to eliminate the fraction:
\[ y^4 = 512y \]
Rearranging gives:
\[ y^4 - 512y = 0 \]
Factoring out \( y \):
\[ y(y^3 - 512) = 0 \]
This gives us \( y = 0 \) or \( y^3 = 512 \). Thus, \( y = 0 \) or \( y = 8 \).

### Step 2: Find Corresponding \( x \) Values
For \( y = 0 \):
\[ x = \frac{0^2}{4} = 0 \]
So one point of intersection is \( (0, 0) \).

For \( y = 8 \):
\[ x = \frac{8^2}{4} = \frac{64}{4} = 16 \]
So the other point of intersection is \( (16, 8) \).

### Step 3: Find the Slopes of the Tangents at the Points of Intersection
To find the slopes of the tangents, we differentiate both equations.

For \( y^2 = 4x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4x) \implies 2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]
At the point \( (16, 8) \):
\[ m_1 = \frac{2}{8} = \frac{1}{4} \]

For \( x^2 = 32y \):
\[ \frac{d}{dx}(x^2) = \frac{d}{dx}(32y) \implies 2x = 32 \frac{dy}{dx} \implies \frac{dy}{dx} = \frac{2x}{32} = \frac{x}{16} \]
At the point \( (16, 8) \):
\[ m_2 = \frac{16}{16} = 1 \]

### Step 4: Calculate the Angle Between the Tangents
The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
Substituting the values we found:
\[ \tan \theta = \left| \frac{1 - \frac{1}{4}}{1 + \left(\frac{1}{4}\right)(1)} \right| = \left| \frac{\frac{3}{4}}{1 + \frac{1}{4}} \right| = \left| \frac{\frac{3}{4}}{\frac{5}{4}} \right| = \frac{3}{5} \]

### Step 5: Find \( \theta \)
Finally, we find \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \]

Thus, the angle at which the parabolas intersect is \( \tan^{-1}\left(\frac{3}{5}\right) \).

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