Question

In the given figure, a circle is centred at (1, 2). The diameter of the circle is

• A. 4
• B. \(2\sqrt{2}\)
• C. \(\sqrt{5}\)
• D. \(2\sqrt{5}\)

Hint:

Always read the question carefully to see whether it asks for the radius or the diameter.
The calculated radius is \(\sqrt{5}\), which is listed as Option (C), but the question asks for the diameter, which is \(2\sqrt{5}\) (Option D).
Do not lose easy marks due to a simple oversight!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The topic of this question is Coordinate Geometry.
We are given a circle plotted on a Cartesian coordinate grid.
The center of the circle is located at the point \(C(1, 2)\).
By observing the circle on the grid, we can see that the circumference of the circle passes exactly through the origin \(O(0, 0)\).
We need to calculate the diameter of this circle.

Step 2: Key Formula or Approach:
1. The radius \(r\) of the circle is the straight-line distance between the center \(C(1, 2)\) and any point on the boundary, which is the origin \(O(0, 0)\).
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] 2. Once the radius \(r\) is calculated, the diameter \(D\) is found by doubling the radius:
\[ \text{Diameter} = 2r \]

Step 3: Detailed Explanation:

• Identify the coordinates of the two key points:
Center of the circle, \(C = (1, 2)\)
Point on the boundary (origin), \(O = (0, 0)\)

• Apply the distance formula to compute the radius \(r = OC\):
\[ r = \sqrt{(1 - 0)^2 + (2 - 0)^2} \] \[ r = \sqrt{1^2 + 2^2} \] \[ r = \sqrt{1 + 4} \] \[ r = \sqrt{5} \text{ units} \]

• Compute the diameter of the circle by doubling the radius:
\[ \text{Diameter} = 2r \] \[ \text{Diameter} = 2\sqrt{5} \text{ units} \]

Step 4: Final Answer:
The diameter of the circle is \(2\sqrt{5}\).
Therefore, the correct option is (D).