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:

The distance from the origin to any point \((x, y)\) is simply \(\sqrt{x^2 + y^2}\). This is a helpful shortcut for coordinate geometry problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Looking at the figure, the circle passes through the origin \((0, 0)\) as the curve touches/crosses the intersection of the axes.
Step 2: Key Formula or Approach:
Radius \(r = \text{distance between center } (1, 2) \text{ and origin } (0, 0)\).
Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Diameter \(D = 2r\).
Step 3: Detailed Explanation:
\[ r = \sqrt{(1 - 0)^2 + (2 - 0)^2} \]
\[ r = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]
Diameter:
\[ D = 2 \times \sqrt{5} = 2\sqrt{5} \]
Step 4: Final Answer:
The diameter of the circle is \(2\sqrt{5}\).