Question

Let \( y^2 = 12x \) be the parabola and \( S \) its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ) = \frac{147}{4} \). Let \( C \) be the circle described by taking \( PQ \) as a diameter. If the equation of the circle \( C \) is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then \( \beta - \alpha \) is equal to: 

Hint:

When dealing with parabolas and focal chords, use the known property of the product of distances and the geometric approach to find the equation of the associated circle.

Answer 1

Correct Answer: 1328

Parabola and Circle Calculation 

We are given the parabola equation:

\[ y^2 = 12x, \quad a = 3 \quad \text{(focus is at } S(3, 0)) \]

The chord PQ satisfies \( (SP)(SQ) = \frac{147}{4} \).

Let \( P(3t^2, 6t) \) and \( Q\left( \frac{9}{4}, -3\sqrt{3} \right) \) be the points on the parabola with parameter \( t \).

From the given condition, we have:

\[ t^2 = \frac{3}{4}, \quad t = \pm \frac{\sqrt{3}}{2} \]

After using the given distances and substituting into the equation of the circle:

\[ (x - 4) \left( x - \frac{9}{4} \right) + (y + 3\sqrt{3})(y - 4\sqrt{3}) = 0 \]

After simplifying, we get:

\[ x^2 + y^2 - \frac{25}{4} - \sqrt{3} y - 27 = 0 \]

Thus, the equation of the circle is obtained and the values of \( \alpha = 400 \), \( \beta = 1728 \).

Finally, we compute:

\[ \beta - \alpha = 1328 \]