This solution will guide you through the process of finding the equation of a circle given a parabola and conditions on its focal chord. We will break down the problem into logical steps, explaining the concepts and formulas used at each stage.

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1. Understanding the Parabola and its Focus

The first step in any problem involving a parabola is to identify its standard form and extract its key parameters.

• Key Concept: The standard form of a parabola with its vertex at the origin and axis along the x-axis is $y^2 = 4ax$.

  • The focus of this parabola is at $S(a, 0)$.
  • The directrix is the line $x = -a$.
  • The latus rectum has length $4a$.

• Given: The equation of the parabola is $y^2 = 12x$.

• Step-by-step:

  1. Compare the given equation $y^2 = 12x$ with the standard form $y^2 = 4ax$.
  2. By direct comparison, we find the value of $4a$: $4a = 12$
  3. Solve for $a$: $a = \frac{12}{4} = 3$
  4. Now, determine the coordinates of the focus $S$. For a parabola $y^2=4ax$, the focus is $(a,0)$. $S = (3, 0)$

• Explanation: The parameter $a$ is fundamental to a parabola as it defines its shape and the positions of its focus and directrix. The focus is a critical point for understanding focal chords and focal distances.

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2. Parametric Representation of Points on the Parabola

To work efficiently with points on a parabola, especially when dealing with chords and distances, it is often advantageous to use parametric coordinates.

• Key Concept: A general point $P$ on the parabola $y^2 = 4ax$ can be represented parametrically as $(at^2, 2at)$, where $t$ is a real parameter.

• Step-by-step:

  1. Substitute the value $a=3$ into the parametric form $(at^2, 2at)$.
  2. Thus, any point on the parabola $y^2=12x$ can be represented as: $P(t) = (3t^2, 6t)$

• Explanation: Using a single parameter $t$ simplifies calculations involving two points on the parabola, as it reduces the number of variables.

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3. Properties of a Focal Chord

A focal chord is a chord of the parabola that passes through its focus. There's a crucial property relating the parameters of its endpoints.

• Key Concept: If $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ are the endpoints of a focal chord of the parabola $y^2=4ax$, then the product of their parameters is $t_1 t_2 = -1$.

• Step-by-step:

  1. Let $P$ be represented by the parameter $t$. So, $P = (3t^2, 6t)$.
  2. According to the focal chord property, the parameter for the other endpoint $Q$ must be $t' = -1/t$.
  3. Substitute $t' = -1/t$ into the parametric form for $Q$: $Q = \left(3\left(-\frac{1}{t}\right)^2, 6\left(-\frac{1}{t}\right)\right) = \left(\frac{3}{t^2}, -\frac{6}{t}\right)$

• Explanation: This property ($t_1 t_2 = -1$) arises from the condition that the slope of the chord $PQ$ must be equal to the slope of the line segment $SP$ (or $SQ$), where $S$ is the focus. It's a fundamental result for parabolas that greatly simplifies calculations involving focal chords.

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4. Focal Distance of a Point on the Parabola

The focal distance is the distance from a point on the parabola to its focus. There's a simple formula for this.

• Key Concept: For a parabola $y^2 = 4ax$ with focus $S(a,0)$, the focal distance of any point $P(x_P, y_P)$ on the parabola is given by $SP = a + x_P$. This property comes directly from the definition of a parabola (a locus of points equidistant from the focus and the directrix $x=-a$).

• Step-by-step:

  1. For point $P(3t^2, 6t)$: $SP = a + x_P = 3 + 3t^2 = 3(1+t^2)$
  2. For point $Q(3/t^2, -6/t)$: $SQ = a + x_Q = 3 + \frac{3}{t^2} = 3\left(1+\frac{1}{t^2}\right)$

• Explanation: Using $a+x_P$ instead of the distance formula $\sqrt{(x_P-a)^2 + y_P^2}$ significantly simplifies calculations and is a common technique in parabola problems.

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5. Using the Given Condition for Focal Distances

We are given a condition involving the product of the focal distances, which we will use to find the value of $t^2$.

• Given: $(SP)(SQ) = \frac{147}{4}$.

• Step-by-step:

  1. Substitute the expressions for $SP$ and $SQ$ derived above: $[3(1+t^2)] \left[3\left(1+\frac{1}{t^2}\right)\right] = \frac{147}{4}$
  2. Simplify the left side: $$$$