Question

Angle between the parabola \[ y^2 = 4(x - 1) \quad \text{and} \quad x^2 + 4(y - 3) = 0 \quad \text{at the common end of their latus rectum is} \]

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{6} \)

Hint:

When finding the angle between two curves, use the formula involving the slopes of the tangents at the point of intersection.

Answer 1

The Correct Option is A

Step 1: Understanding the geometrical configuration.
We are asked to find the angle between the parabola \( y^2 = 4(x - 1) \) and the equation \( x^2 + 4(y - 3) = 0 \), at the common end of their latus rectum. First, let's rewrite the equations of both curves.

Step 2: Equation of the parabola.
The equation of the parabola is:
\[ y^2 = 4(x - 1), \]
which represents a parabola opening towards the right. The vertex is at \( (1, 0) \), and the focus is at \( (2, 0) \).

Step 3: Equation of the second curve.
The second curve is:
\[ x^2 + 4(y - 3) = 0 \quad \Rightarrow \quad x^2 = -4(y - 3) \quad \Rightarrow \quad y = 3 - \frac{x^2}{4}. \]
This is the equation of a downward-opening parabola with vertex at \( (0, 3) \) and focus at \( \left( 0, 3 + \frac{1}{4} \right) \).

Step 4: Finding the angle between the tangents.
The general formula for the angle \( \theta \) between two curves at a point is:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|, \]
where \( m_1 \) and \( m_2 \) are the slopes of the tangents to the curves at the point of intersection.
The slope of the tangent to the parabola \( y^2 = 4(x - 1) \) is given by:
\[ m_1 = \frac{dy}{dx} = \frac{2}{4} = \frac{1}{2}. \]
The slope of the tangent to the curve \( y = 3 - \frac{x^2}{4} \) is:
\[ m_2 = \frac{dy}{dx} = -\frac{x}{4}. \]
At the common end of the latus rectum, \( x = 2 \) (for the parabola), so:
\[ m_2 = -\frac{2}{4} = -\frac{1}{2}. \]

Step 5: Calculating the angle.
Now, substitute \( m_1 = \frac{1}{2} \) and \( m_2 = -\frac{1}{2} \) into the angle formula:
\[ \tan \theta = \left| \frac{\frac{1}{2} - (-\frac{1}{2})}{1 + \frac{1}{2} \times -\frac{1}{2}} \right| = \left| \frac{1}{1 - \frac{1}{4}} \right| = \left| \frac{1}{\frac{3}{4}} \right| = \frac{4}{3}. \]
Thus, the angle is:
\[ \theta = \tan^{-1} \left( \frac{4}{3} \right) = \frac{\pi}{4}. \]
Final Answer:
The angle between the two parabolas at the common end of their latus rectum is: \[ \boxed{\frac{\pi}{4}}. \]