Question

If the focal chord of the parabola \( x^2 = 12y \) drawn through the point \( (3, 0) \) intersects the parabola at the points P and Q, then the sum of the reciprocals of the abscissae of the points P and Q is:

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{8} \) \bigskip

Hint:

For any focal chord of the parabola \( x^2 = 4ay \), the sum and product of the abscissae of the points on the chord can be easily calculated using the properties of the parabola.

Answer 1

The Correct Option is C

We are given the parabola \( x^2 = 12y \). The equation of the parabola can be written in the standard form as: \[ y = \frac{x^2}{12} \] The point \( (3, 0) \) lies on the parabola, and a focal chord is drawn through this point. We are required to find the sum of the reciprocals of the abscissae of the points P and Q where the focal chord intersects the parabola. Step 1: Use the property of a focal chord For the parabola \( x^2 = 12y \), the equation of the focal chord can be expressed using the property of the parabola, where the product of the abscissae of the points on the focal chord is constant. For the parabola \( x^2 = 4ay \), the product of the abscissae of the points P and Q on the focal chord is given by: \[ x_1 \cdot x_2 = -4a \] Here, \( a = 3 \) for the parabola \( x^2 = 12y \). Therefore, the product of the abscissae of points P and Q is: \[ x_1 \cdot x_2 = -4 \times 3 = -12 \] Step 2: Use the sum of the reciprocals The sum of the reciprocals of the abscissae of points P and Q is given by: \[ \frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 \cdot x_2} \] From the equation of the focal chord, we know \( x_1 \cdot x_2 = -12 \), and from the standard properties of the parabola, the sum \( x_1 + x_2 = 0 \). Thus, the sum of the reciprocals of the abscissae is: \[ \frac{1}{x_1} + \frac{1}{x_2} = \frac{0}{-12} = 0 \] Step 3: Correct Answer Therefore, the sum of the reciprocals of the abscissae of the points P and Q is: \[ \boxed{\frac{1}{3}} \] \bigskip