Question

PSQ is a focal chord of the parabola \[ y^2=12x. \] \(A\) and \(B\) are respectively the feet of the perpendiculars drawn from \(P\) and \(Q\) on the directrix of the parabola. If the length of \(AB\) is \(7\sqrt3\) and \[ P=(3t^2,6t), \qquad (0<t<1), \] then \(t=\)

• A. \[ \frac{2}{\sqrt3} \]
• B. \[ \frac{\sqrt3}{2} \]
• C. \[ \frac{\sqrt2}{3} \]
• D. \[ \frac{\sqrt5}{4} \] \bigskip

Hint:

For the parabola \[ y^2=4ax, \] the ends of a focal chord have parameters \(t\) and \(-1/t\). This property is frequently used in JEE and entrance examinations.

Answer 1

The Correct Option is B

Concept: For the parabola \[ y^2=4ax, \] a parametric point is \[ (at^2,\,2at). \] If \(P(at_1^2,2at_1)\) and \(Q(at_2^2,2at_2)\) are the ends of a focal chord, then \[ t_1t_2=-1. \] The directrix is \[ x=-a. \]

Step 1: Identify the parameter and the second end of the focal chord. Given \[ y^2=12x, \] so \[ 4a=12 \quad\Rightarrow\quad a=3. \] The point \[ P=(3t^2,6t) \] corresponds to parameter \(t\). Since \(PSQ\) is a focal chord, \[ t\cdot t_2=-1. \] Hence \[ t_2=-\frac1t. \] Therefore, \[ Q= \left( \frac{3}{t^2}, -\frac{6}{t} \right). \]

Step 2: Find the coordinates of \(A\) and \(B\). The directrix is \[ x=-3. \] The feet of the perpendiculars from \(P\) and \(Q\) onto the directrix are \[ A=(-3,6t), \] \[ B=\left(-3,-\frac6t\right). \]

Step 3: Compute \(AB\). Since both points lie on the vertical line \(x=-3\), \[ AB = \left| 6t-\left(-\frac6t\right) \right|. \] \[ = 6\left(t+\frac1t\right). \] Given \[ AB=7\sqrt3, \] hence \[ 6\left(t+\frac1t\right)=7\sqrt3. \] \[ t+\frac1t=\frac{7\sqrt3}{6}. \]

Step 4: Solve for \(t\). Multiplying by \(t\), \[ t^2+1=\frac{7\sqrt3}{6}t. \] \[ 6t^2-7\sqrt3\,t+6=0. \] Using the quadratic formula, \[ t = \frac{7\sqrt3\pm\sqrt{147-144}}{12}. \] \[ = \frac{7\sqrt3\pm\sqrt3}{12}. \] \[ = \frac{\sqrt3(7\pm1)}{12}. \] Thus, \[ t=\frac{2\sqrt3}{3} \] or \[ t=\frac{\sqrt3}{2}. \] Since \[ 0<t<1, \] we choose \[ t=\frac{\sqrt3}{2}. \]

Step 5: Write the final answer. \[ \boxed{\frac{\sqrt3}{2}} \]