Let `C :y = x^(2) -3, D : y = kx^(2)` be two parabolas and
`L_(1) : x = a , L_(2) : x = 1 (a ne 0)` be two straight lines.
If `a gt 0`, the angle subtended by the chord AB at the vertex of the parabola C is

Options

A. A) `tan^(-1)((5)/(7))`
B. B) `tan^(-1)((5)/(2))`
C. C) `tan^(-1)[1/(2)`]
D. D)`tan^(-1)((1)/(8))`

Answer

To solve the problem, we need to find the angle subtended by the chord AB at the vertex of the parabola \( C: y = x^2 - 3 \). Let's break this down step by step.

### Step 1: Identify the equations of the parabolas and lines
The equations of the parabolas are:
- \( C: y = x^2 - 3 \)
- \( D: y = kx^2 \)

The equations of the lines are:
- \( L_1: x = a \)
- \( L_2: x = 1 \) (where \( a \neq 0 \))

### Step 2: Find the points of intersection of the lines with the parabolas
For line \( L_1: x = a \):
1. Substitute \( x = a \) into the equation of parabola \( C \):
\[ y = a^2 - 3 \]
Thus, point \( A \) is \( (a, a^2 - 3) \).

2. Substitute \( x = a \) into the equation of parabola \( D \):
\[ y = ka^2 \]
Thus, point \( B \) is \( (a, ka^2) \).

For line \( L_2: x = 1 \):
1. Substitute \( x = 1 \) into the equation of parabola \( C \):
\[ y = 1^2 - 3 = -2 \]
Thus, point \( C \) is \( (1, -2) \).

2. Substitute \( x = 1 \) into the equation of parabola \( D \):
\[ y = k(1^2) = k \]
Thus, point \( D \) is \( (1, k) \).

### Step 3: Find the coordinates of points A and B
From the previous calculations:
- Point \( A \) is \( (a, a^2 - 3) \)
- Point \( B \) is \( (1, k) \)

### Step 4: Find the vertex of parabola \( C \)
The vertex of parabola \( C: y = x^2 - 3 \) is at the point \( (0, -3) \).

### Step 5: Calculate the slopes of lines OA and OB
1. **Slope of OA**:
\[ \text{slope of OA} = \frac{(a^2 - 3) - (-3)}{a - 0} = \frac{a^2}{a} = a \]

2. **Slope of OB**:
\[ \text{slope of OB} = \frac{k - (-3)}{1 - 0} = k + 3 \]

### Step 6: Find the angle between lines OA and OB
The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting \( m_1 = a \) and \( m_2 = k + 3 \):
\[ \tan \theta = \left| \frac{a - (k + 3)}{1 + a(k + 3)} \right| \]

### Step 7: Conclusion
The angle subtended by the chord AB at the vertex of the parabola C is given by:
\[ \theta = \tan^{-1}\left( \frac{a - (k + 3)}{1 + a(k + 3)} \right) \]