© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the Parabola and its Latus Rectum
The parabola is given by $y^2 = 4x$. In this standard form ($y^2 = 4ax$), we have $a = 1$. For a parabola $y^2 = 4ax$, the latus rectum is the line segment perpendicular to the axis of the parabola, passing through the focus. Its equation is $x = a$, so here the latus rectum has the equation
$x = 1$.
Endpoints of this latus rectum are at $(1, 2)$ and $(1, -2)$. Hence, its length is $4$.
Step 2: Using the Common Chord Property for the Circles
We are told that this line $x = 1$ (the latus rectum) is the common chord to both circles $C_1$ and $C_2$, each of which has radius $R = 2\sqrt{5}$. The chord length is $4$, as found from the endpoints $(1,2)$ and $(1,-2)$.
Step 3: Relate the Chord Length to the Radius and Perpendicular Distance
For a circle of radius $R$, if a chord has length $c$, and its perpendicular distance from the center is $d$, then
$c = 2 \sqrt{R^2 - d^2}.
Here, $c = 4$ and $R = 2\sqrt{5}$. Substituting these values:
$4 = 2 \sqrt{(2\sqrt{5})^2 - d^2} \; \Longrightarrow \; 2 = \sqrt{20 - d^2}.
Squaring both sides:
$4 = 20 - d^2 \; \Longrightarrow \; d^2 = 16 \; \Longrightarrow \; d = 4.
Thus, the perpendicular distance of each circle's center to the line $x = 1$ is $4$.
Step 4: Determine the Position of the Centers and Find the Distance
Because the chord is vertical ($x=1$), each center must be $4$ units horizontally away from this line. Therefore, if one center is at $x = 1 + 4 = 5$ and the other is at $x = 1 - 4 = -3$, both lying on the same horizontal (say the $x$-axis for simplicity), then the distance between the two centers is
$(5) - (-3) = 8.
Answer
The distance between the centres of the circles is $\boxed{8}$.
