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:

Be careful to distinguish between radius and diameter.
The radius is \(\sqrt{5}\), which is listed as Option (C), but the question asks for the diameter, which is \(2\sqrt{5}\) (Option D).
Always read the final line of the question carefully!

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 system.
The center of this circle is located at the point \(C(1, 2)\).
By observing the graph, we can see that the circumference of the circle passes through the origin \(O(0, 0)\).
We need to calculate the diameter of this circle.

Step 2: Key Formula or Approach:
- Calculate the radius \(r\) of the circle, which is the distance between the center \(C(1, 2)\) and the point \(O(0, 0)\) on its boundary.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] - The diameter of a circle is twice its radius:
\[ \text{Diameter} = 2r \]

Step 3: Detailed Explanation:

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

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

• Calculate the diameter:
\[ \text{Diameter} = 2r = 2\sqrt{5} \text{ units} \]

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