Question

Let the latus ractum of the parabola $y^{2}=4x$ be the common chord to the circles $C _{1}$ and $C _{2}$ each of them having radius $2 \sqrt{5}$. Then, the distance between the centres of the circles $C _{1}$ and $C _{2}$ is:

• A. $8$
• B. $4 \sqrt{5}$
• C. $12$
• D. $8 \sqrt{5}$

Answer 1

The Correct Option is A

To solve this problem, we need to find the distance between the centers of two circles \(C_1\) and \(C_2\), having the same radius and a common chord which is the latus rectum of the parabola \(y^2 = 4x\).

Step 1: Understand the latus rectum of the parabola.

The given parabola is \(y^2 = 4x\). The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry, passing through the focus, and whose endpoints lie on the parabola. For this parabola, \(y^2 = 4ax\), where \(a=1\).

The length of the latus rectum for a parabola \(y^2 = 4ax\) is given by \(4a\). Therefore, here it is \(4 \times 1 = 4\).

The equation of the latus rectum is \(x = a\). Here, it is \(x = 1\).

Step 2: Understand the meaning of the common chord.

The common chord of the two circles is along the line \(x = 1\). Let the centers of the circles be \((h_1, k_1)\) and \((h_2, k_2)\). The circles have a radius of \(2\sqrt{5}\).

Step 3: Determine conditions for the common chord.

Since the line \(x = 1\) is the common chord, the direct distances from the circle centers to the chord need to relate to their radii:

The distance from the center of a circle \((h, k)\) to the line \(x = 1\) is calculated as:

\(|h - 1| \le r = 2\sqrt{5}\)

If the centers lie symmetrically around \(x = 1\) on either side of this line such that \(h_1 - 1 = -(h_2 - 1)\), the distance between the centers \(d\) is twice the horizontal distance from the centers to the line:

Step 4: Calculate the distance between the centers.

Therefore, the total horizontal distance involved is:

\(|h_1 - h_2| = 2|h_1 - 1|\)

Since \(|h_1 - 1| = 2\sqrt{5}\) and \(h_1 - h_2 = 4\) (considering symmetry),

\(D = d = 8\)

Conclusion: Thus, the distance between the centers of the two circles is \(8\).