Question

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on \(3x + 2y + 2 = 0\). Then the length of the chord of the circle C, whose midpoint is (1, 2), is:

• A. \(\sqrt{3}\)
• B. \(2\sqrt{3}\)
• C. \(4\sqrt{2}\)
• D. 2\(\sqrt{2}\)

Hint:

For chord calculations, using the radius relation with known points simplifies the calculation efficiently.

Answer 1

The Correct Option is B

Step 1: Identifying the equation of the circle. Given that the circle passes through points \((4, 2)\) and \((0, 2)\), the general form of the circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Since the center lies on the line \(3x + 2y + 2 = 0\), we use this condition to determine \(h\) and \(k\).

Step 2: Finding the radius. From the midpoint condition, and computing distances: \[ ON = \sqrt{(h - 1)^2 + (k - 2)^2} = \sqrt{37} \]

Step 3: Finding the chord length. Using the chord length formula: \[ \text{Chord Length} = 2\sqrt{r^2 - (ON)^2} = 2\sqrt{40 - 37} = 2\sqrt{3} \]