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

\includegraphics[width=0.5\linewidth]{4.png}
Step 1: Find intersection points. The given equations are: \[ 3x - 2y + 12 = 0, \quad 4y = 3x^2. \] Substituting \( y = \frac{3x^2}{4} \) into the line equation: \[ 3x - 2\left(\frac{3x^2}{4}\right) + 12 = 0. \] \[ 3x - \frac{6x^2}{4} + 12 = 0 \Rightarrow x^2 - 2x - 8 = 0. \] Solving for \( x \), we get: \[ x = -2, 4. \] Step 2: Find slopes. The slopes of the lines from the vertex \( (0,0) \) to the intersection points are: \[ m_{OA} = \frac{3}{-2} = -\frac{3}{2}, \quad m_{OB} = \frac{12}{4} = 3. \] Step 3: Compute the angle. Using the formula: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] \[ \tan \theta = \left| \frac{-\frac{3}{2} - 3}{1 + \left(-\frac{3}{2} \times 3\right)} \right| \] \[ = \left| \frac{-\frac{3}{2} - \frac{6}{2}}{1 - \frac{9}{2}} \right| = \left| \frac{-\frac{9}{2}}{1 - \frac{9}{2}} \right| = \frac{9}{7}. \] Thus, \[ \theta = \tan^{-1} \left(\frac{9}{7} \right). \]