Step 1: Understanding the Question:
The angle between two curves is the angle between their tangents at the point of intersection. We first need to find the common endpoint of their latus recta.
Step 2: Key Formula or Approach:
1. Find the point of intersection.
2. Find the slopes \( m_1 \) and \( m_2 \) of the tangents at that point.
3. Use \( \tan \theta = | \frac{m_1 - m_2}{1 + m_1 m_2} | \).
Step 3: Detailed Explanation:
Parabola 1: \( y^2 = 4(x-1) \). This is of form \( Y^2 = 4aX \) where \( a = 1 \), vertex is \( (1,0) \).
Focus is \( (1+1, 0) = (2,0) \). Latus rectum is the line \( x=2 \). Endpoints are \( (2, 2) \) and \( (2, -2) \).
Parabola 2: \( x^2 = -4(y-3) \). This is of form \( X^2 = -4aY \) where \( a = 1 \), vertex is \( (0,3) \).
Focus is \( (0, 3-1) = (0,2) \). Latus rectum is the line \( y=2 \). Endpoints are \( (2, 2) \) and \( (-2, 2) \).
The common endpoint is \( (2,2) \).
Slope \( m_1 \) for \( y^2 = 4x - 4 \):
Differentiating: \( 2y \frac{dy}{dx} = 4 \Rightarrow \frac{dy}{dx} = \frac{2}{y} \).
At \( (2,2) \), \( m_1 = \frac{2}{2} = 1 \).
Slope \( m_2 \) for \( x^2 = -4y + 12 \):
Differentiating: \( 2x = -4 \frac{dy}{dx} \Rightarrow \frac{dy}{dx} = -\frac{x}{2} \).
At \( (2,2) \), \( m_2 = -\frac{2}{2} = -1 \).
Since \( m_1 \cdot m_2 = (1)(-1) = -1 \), the tangents are perpendicular.
The angle \( \theta = \frac{\pi}{2} \).
Step 4: Final Answer:
The angle between the curves is \( \pi/2 \).