Now we equate the calculated product (SP)(SQ) to the given value (147)/(4) and solve for t_1². This leads to a quadratic equation in t_1². Factoring the quadratic equation 12(t_1²)² - 25t_1² + 12 = 0, we find two possible values for t_1²: (3)/(4) or (4)/(3). Let's choose t_1² = (3)/(4).

Now we equate the calculated product (SP)(SQ) to the given value (147)/(4) and solve for t_1². This leads to a quadratic equation in t_1². Factoring the quadratic equation 12(t_1²)² - 25t_1² + 12 = 0, we find two possible values for t_1²: (3)/(4) or (4)/(3). Let's choose t_1² = (3)/(4).

Using t_1² = (3)/(4), we get t_1 = (√3)/(2). Then t_2 = -(1)/(t_1) = -(2)/(√3). Now we can find the coordinates of points P and Q using the parametric form (at², 2at). Substituting a=3 and the values of t_1 and t_2, we get P((9)/(4), 3√3) and Q(4, -4√3).

Using t_1² = (3)/(4), we get t_1 = (√3)/(2). Then t_2 = -(1)/(t_1) = -(2)/(√3). Now we can find the coordinates of points P and Q using the parametric form (at², 2at). Substituting a=3 and the values of t_1 and t_2, we get P((9)/(4), 3√3) and Q(4, -4√3).

Comparing our derived equation 64x² + 64y² - 400x + 64√3y - 1728 = 0 with the given equation 64x² + 64y² - α x - 64√3y = β, we can identify the values of α and β. We find that α = 400 and β = 1728. Finally, we calculate β - α = 1728 - 400 = 1328.

Comparing our derived equation 64x² + 64y² - 400x + 64√3y - 1728 = 0 with the given equation 64x² + 64y² - α x - 64√3y = β, we can identify the values of α and β. We find that α = 400 and β = 1728. Finally, we calculate β - α = 1728 - 400 = 1328.

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

Answer: 1328

Step-by-step solution

Step 1: Identify Parabola Properties and Focal Chord Length

First, we identify the properties of the given parabola $y^2 = 12x$ . Comparing it with the standard form $y^2 = 4ax$ , we find that $4a = 12$ , which means $a = 3$ . The focus $S$ of the parabola is at $(a, 0)$ , so $S = (3, 0)$ . For a focal chord $PQ$ with endpoints $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ , the lengths $SP$ and $SQ$ are given by $a(1+t_1^2)$ and $a(1+t_2^2)$ respectively. Since $PQ$ is a focal chord, the product of the parameters $t_1t_2 = -1$ .

Step 2: Calculate Product of Focal Distances

We are given that $(SP)(SQ) = \frac{147}{4}$ . We substitute the expressions for $SP$ and $SQ$ and use the relation $t_2 = -1/t_1$ . This simplifies the product to $9(1+t_1^2)(1+1/t_1^2)$ . Expanding this, we get $9(2+t_1^2+1/t_1^2)$ .

Step 3: Solve for $t_1^2$

Now we equate the calculated product $(SP)(SQ)$ to the given value $\frac{147}{4}$ and solve for $t_1^2$ . This leads to a quadratic equation in $t_1^2$ . Factoring the quadratic equation $12(t_1^2)^2 - 25t_1^2 + 12 = 0$ , we find two possible values for $t_1^2$ : $\frac{3}{4}$ or $\frac{4}{3}$ . Let's choose $t_1^2 = \frac{3}{4}$ .

Step 4: Determine Coordinates of P and Q

Using $t_1^2 = \frac{3}{4}$ , we get $t_1 = \frac{\sqrt{3}}{2}$ . Then $t_2 = -\frac{1}{t_1} = -\frac{2}{\sqrt{3}}$ . Now we can find the coordinates of points $P$ and $Q$ using the parametric form $(at^2, 2at)$ . Substituting $a=3$ and the values of $t_1$ and $t_2$ , we get $P(\frac{9}{4}, 3\sqrt{3})$ and $Q(4, -4\sqrt{3})$ .

Step 5: Find the Equation of Circle C

The circle $C$ is described taking $PQ$ as a diameter. The equation of a circle with endpoints of a diameter $(x_1, y_1)$ and $(x_2, y_2)$ is given by $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$ . Substituting the coordinates of $P$ and $Q$ , we expand and simplify the equation. Multiplying by 64 to match the given form, we get $64x^2 + 64y^2 - 400x + 64\sqrt{3}y - 1728 = 0$ .

Step 6: Compare and Calculate $\beta - \alpha$

Comparing our derived equation $64x^2 + 64y^2 - 400x + 64\sqrt{3}y - 1728 = 0$ with the given equation $64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta$ , we can identify the values of $\alpha$ and $\beta$ . We find that $\alpha = 400$ and $\beta = 1728$ . Finally, we calculate $\beta - \alpha = 1728 - 400 = 1328$ .

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