Question

If the line \( 3x - 2y + 12 = 0 \) intersects the parabola \( 4y = 3x^2 \) at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to:

• A. \( \tan^{-1} \left(\frac{11}{9} \right) \)
• B. \( \frac{\pi}{2} - \tan^{-1} \left(\frac{3}{2} \right) \)
• C. \( \tan^{-1} \left(\frac{4}{5} \right) \)
• D. \( \tan^{-1} \left(\frac{9}{7} \right) \)

Hint:

For angles subtended by chords at the vertex of a parabola, use the tangent formula to determine the required angle.

Answer 1

The Correct Option is D

To determine the angle subtended by line segment \( AB \) at the parabola's vertex, we first find the intersection points \( A \) and \( B \) of the line and the parabola.

1. Rearrange the equations for the line and parabola for simpler calculations:
   The line is \( 3x - 2y + 12 = 0 \), which can be rewritten as \( y = \frac{3}{2}x + 6 \).
   The parabola's equation is \( 4y = 3x^2 \), or \( y = \frac{3}{4}x^2 \).
2. Equate the expressions for \( y \) to find the \( x \)-coordinates of points \( A \) and \( B \):
   \(\frac{3}{4}x^2 = \frac{3}{2}x + 6\)
   Multiply by 4 to clear fractions:
   \(3x^2 = 6x + 24\)
   Rearrange into a quadratic equation:
   \(3x^2 - 6x - 24 = 0\).
3. Simplify and solve the quadratic equation:
   Divide by 3:
   \(x^2 - 2x - 8 = 0\).
   Factorize:
   \((x - 4)(x + 2) = 0\).
   The solutions are \( x = 4 \) and \( x = -2 \).
4. Substitute the \( x \)-values back into the parabola's equation to find the corresponding \( y \)-coordinates:
   For \( x = 4 \), \( y = \frac{3}{4}(4)^2 = 12 \).
   For \( x = -2 \), \( y = \frac{3}{4}(-2)^2 = 3 \).
5. The intersection points are \( A(4, 12) \) and \( B(-2, 3) \).
6. Identify the vertex of the parabola. For \( y = \frac{3}{4}x^2 \), the vertex is at \( V(0, 0) \).
7. Calculate the slopes of the lines connecting the vertex to points \( A \) and \( B \):
   • Slope of \( VA \): \( m_1 = \frac{12 - 0}{4 - 0} = 3 \).
   • Slope of \( VB \): \( m_2 = \frac{3 - 0}{-2 - 0} = -\frac{3}{2} \).
8. Use the formula for the angle \( \theta \) between two lines: \( \theta = \tan^{-1}\left( \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right) \).
   Substitute the slopes:
   \(\theta = \tan^{-1}\left( \left| \frac{3 - \left(-\frac{3}{2}\right)}{1 + 3 \left(-\frac{3}{2}\right)} \right| \right)\).
   Simplify:
   \(\theta = \tan^{-1}\left( \left| \frac{\frac{9}{2}}{1 - \frac{9}{2}} \right| \right) = \tan^{-1}\left( \left| \frac{\frac{9}{2}}{-\frac{7}{2}} \right| \right) = \tan^{-1}\left( \left| -\frac{9}{7} \right| \right) = \tan^{-1}\left( \frac{9}{7} \right)\).

The angle subtended is \( \tan^{-1} \left(\frac{9}{7} \right) \).