This problem is a fantastic blend of concepts from coordinate geometry, specifically parabolas and circles, combined with fundamental Euclidean geometry principles like the Pythagorean theorem and properties of chords. To solve it effectively, we will break it down into manageable steps.

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Key Concepts and Formulas

Before diving into the solution, let's review the essential concepts we'll be using:

1. Parabola $y^2 = 4ax$:

   • This is the standard form of a parabola with its vertex at the origin $(0,0)$ and its axis of symmetry along the positive x-axis.
   • The parameter '$a$' defines the parabola's shape and focal length.
   • Focus: The focus is located at the point $(a,0)$.
   • Latus Rectum: This is a special chord of the parabola that passes through the focus and is perpendicular to the axis of the parabola.
     • Its equation is $x=a$.
     • Its endpoints are $(a, 2a)$ and $(a, -2a)$.
     • Its length is $4a$.

2. Circles and Common Chord:

   • When two circles intersect at two distinct points, the line segment connecting these two points is called their common chord.
   • A crucial geometric property is that the line segment joining the centers of the two circles is always perpendicular to their common chord and bisects it. This property is fundamental for forming right-angled triangles and solving problems involving intersecting circles.
   • Radius ($R$): The distance from the center of a circle to any point on its circumference.
   • Distance from Center to Chord: If a chord of length $L$ is within a circle of radius $R$, and $d$ is the perpendicular distance from the center to the chord, then a right-angled triangle is formed. The sides of this triangle are $d$, $L/2$, and the hypotenuse $R$.

3. Pythagorean Theorem:

   • In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. If the sides are $A$, $B$, and the hypotenuse is $C$, then $A^2 + B^2 = C^2$.

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Step-by-Step Solution

Step 1: Determine the Length of the Latus Rectum of the Parabola

• Goal: The problem states that the latus rectum of the given parabola is the common chord for the two circles. Our first step is to find the length of this latus rectum.
• Given Parabola: We are provided with the equation of the parabola: $y^2 = 4x$
• Identify 'a': We compare this equation with the standard form of a parabola opening to the right, which is $y^2 = 4ax$. By comparing the coefficients of $x$: $4a = 4$ Solving for $a$: $a = 1$
  • Why this step? The parameter '$a$' is crucial because the length of the latus rectum is directly determined by $4a$. Finding 'a' allows us to calculate this length.
• Calculate Length of Latus Rectum: The length of the latus rectum for a parabola $y^2 = 4ax$ is $4a$. Substituting the value $a=1$: $\text{Length of Latus Rectum} = 4(1) = 4$
• Identify the Common Chord: As stated in the problem, this latus rectum is the common chord for circles $C_1$ and $C_2$. Therefore, the length of the common chord is $4$.
  • Why this step? This establishes a key piece of information—the length of the common chord—which is essential for subsequent calculations involving the circles. Let's denote this common chord as $PQ$, so $PQ = 4$.

Step 2: Understand the Geometry of the Circles and their Common Chord

• Goal: Set up the geometric configuration of the circles, their centers, and the common chord to prepare for applying the Pythagorean theorem.
• Let $O_1$ and $O_2$ be the centers of circles $C_1$ and $C_2$, respectively.
• The radius of each circle is given as $R = 2\sqrt{5}$.
• The length of the common chord $PQ$ is $4$, as determined in Step 1.
• Key Geometric Property: The line segment joining the centers $O_1O_2$ has a specific relationship with the common chord $PQ$. It is always perpendicular to $PQ$ and bisects $PQ$.
  • Why this property is important? This property allows us to divide the common chord into two equal halves and, more importantly, forms a right-angled triangle when we connect a center to an endpoint of the chord and to its midpoint. This right-angled triangle is where the Pythagorean theorem will be applied.
• Let $M$ be the midpoint of the common chord $PQ$. Then, $PM = MQ = \frac{PQ}{2}$.
• Calculate Half-Length of Chord: $PM = \frac{4}{2} = 2$
  • Why this step? $PM$ represents half the length of the common chord, which will be one of the legs of the right-angled triangle we are about to form.

Step 3: Apply the Pythagorean Theorem to find the Distance from a Circle's Center to the Common Chord

• Goal: Calculate the perpendicular distance from the center of one circle to the common chord.
• Consider circle $C_1$ with center $O_1$ and radius $R$.
• We can form a right-angled triangle, $\triangle O_1MP$, by connecting:
  • The center $O_1$ to one endpoint of the common chord, say $P$. This line segment $O_1P$ is the radius of the circle, so $O_1P = R = 2\sqrt{5}$. This will be the hypotenuse of our right triangle.
  • The center $O_1$ to the midpoint $M$ of the common chord. This line segment $O_1M$ represents the perpendicular distance from the center to the chord. This will be one leg of the triangle.
  • The midpoint $M$ to the endpoint $P$. This line segment $PM$ is half the length of the common chord, which we found to be $2$. This will be the other leg.
• Since the line $O_1O_2$ is perpendicular to $PQ$ at $M$, the angle $\angle O_1MP$ is $90^\circ$. Thus, $\triangle O_1MP$ is indeed a right-angled triangle.
• Apply Pythagorean Theorem: According to the theorem, for $\triangle O_1MP$: $(O_1P)^2 = (PM)^2 + (O_1M)^2$ Substitute the known values: $(2\sqrt{5})^2 = (2)^2 + (O_1M)^2$ Calculate the squares carefully: $(2^2 \times (\sqrt{5})^2) = 4 + (O_1M)^2$ $(4 \times 5) = 4 + (O_1M)^2$ $20 = 4 + (O_1M)^2$
  • Why this step? The Pythagorean theorem is the tool that allows us to find the unknown distance $O_1M$ by relating the known radius (hypotenuse) and half-chord length (one leg).
• Solve for $O_1M$: $(O_1M)^2 = 20 - 4$ $(O_1M)^2 = 16$ Take the positive square root (since distance must be a positive value): $O_1M = \sqrt{16}$ $O_1M = 4$ This value, $O_1M=4$, is the perpendicular distance from the center of circle $C_1$ to the common chord.

Step 4: Calculate the Distance Between the Centers of the Circles

• Goal: Determine the total distance between $O_1$ and $O_2$.
• The line segment connecting the centers, $O_1O_2$, passes through the midpoint $M$ of the common chord.
• Since both circles $C_1$ and $C_2$ have the same radius ($2\sqrt{5}$), they are identical. This implies that their centers must be equidistant from the common chord. Therefore, the distance from $O_2$ to $M$ is the same as $O_1M$: $MO_2 = O_1M = 4$
  • Why this step? The symmetry of the problem (identical circles) ensures that the common chord is positioned symmetrically with respect to both centers. The midpoint $M$ of the common chord lies on the line segment $O_1O_2$.
• Total Distance Between Centers: The total distance between the centers $O_1$ and $O_2$ is the sum of the distances $O_1M$ and $MO_2$. $O_1O_2 = O_1M + MO_2$ $O_1O_2 = 4 + 4$ $O_1O_2 = 8$
  • Why this step? This is the final value requested by the question, obtained by summing the individual distances from each circle's center to the common chord along the line connecting the centers.

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Tips and Common Mistakes to Avoid

• Visualize with a Sketch: Always start by drawing a simple diagram. This helps immensely in understanding the spatial relationships between the parabola, circles, centers, radii, and the common chord. It makes it easier to identify the right-angled triangle.
• Parabola Parameters: For a parabola $y^2 = 4ax$, remember that the latus rectum is the line $x=a$ and its length is $4a$. Don't confuse the parameter 'a' with the length of the latus rectum, which is $4a$.
• Common Chord Properties are Crucial: The property that the line joining the centers of two intersecting circles is perpendicular to and bisects their common chord is fundamental. If you forget this, you won't be able to form the right-angled triangle correctly.
• Pythagorean Theorem Application: Ensure you correctly identify the hypotenuse (always the radius when dealing with a center, chord, and perpendicular distance) and the legs of the right triangle. A common error is mixing them up.
• Calculation Errors: Be meticulous with squaring terms involving square roots. For instance, $(2\sqrt{5})^2 = 2^2 \times (\sqrt{5})^2 = 4 \times 5 = 20$. A frequent mistake is to forget to square the coefficient (the '2' in this case), leading to $2 \times 5 = 10$.

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Summary and Key Takeaway

This problem beautifully illustrates how to integrate concepts from different areas of coordinate geometry. We began by extracting crucial information about the common chord from the parabola's equation, finding its length ($4a$) to be $4$. This length then became the basis for our calculations involving the two circles. By leveraging the critical geometric property that the line connecting the centers of two intersecting circles is perpendicular to and bisects their common chord, we were able to construct a right-angled triangle. Using the given radius ($2\sqrt{5}$) and half the common chord length ($2$), the Pythagorean theorem allowed us to calculate the perpendicular distance from each circle's center to the