Question

If the ratio of y-ordinates of points on the parabola \( y^2 = 12x \) is 1 : 2 and the length of the chord joining these points is \( 3\sqrt{13} \), then find the angle subtended by the chord at the focus of the parabola:

• A. \( \tan^{-1} \frac{1}{2} \)
• B. \( \tan^{-1} \frac{3}{4} \)
• C. \( \tan^{-1} \frac{2}{3} \)
• D. \( \tan^{-1} \frac{1}{4} \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For the parabola \( y^2 = 4ax \), \( 4a = 12 \implies a = 3 \). Points on the parabola can be represented as \( (at^2, 2at) \). We use the given ratio and chord length to find the parameters \( t_1 \) and \( t_2 \).

Step 2: Key Formula or Approach:
1. Let points be \( P(3t_1^2, 6t_1) \) and \( Q(3t_2^2, 6t_2) \). 2. Given \( \frac{6t_1}{6t_2} = \frac{1}{2} \implies t_2 = 2t_1 \). 3. Distance formula: \( PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = 3\sqrt{13} \).

Step 3: Detailed Explanation:
1. Substitute coordinates: \( \sqrt{(3(2t_1)^2 - 3t_1^2)^2 + (12t_1 - 6t_1)^2} = 3\sqrt{13} \). 2. \( \sqrt{(9t_1^2)^2 + (6t_1)^2} = 3\sqrt{13} \implies 81t_1^4 + 36t_1^2 = 117 \). 3. Let \( u = t_1^2 \): \( 9u^2 + 4u - 13 = 0 \implies (u-1)(9u+13) = 0 \). 4. Since \( t_1^2 \) is positive, \( t_1^2 = 1 \). Let \( t_1 = 1, t_2 = 2 \). 5. Points: \( P(3, 6) \) and \( Q(12, 12) \). Focus \( S(3, 0) \). 6. Slopes \( m_{SP} = \text{undefined (vertical)} \) and \( m_{SQ} = \frac{12-0}{12-3} = \frac{12}{9} = \frac{4}{3} \). 7. Angle \( \theta \) between vertical line and line with slope 4/3 is \( \tan^{-1}(3/4) \).

Step 4: Final Answer:
The angle subtended at the focus is \( \tan^{-1} \frac{3}{4} \).