AB is a chord of `x^2 + y^2 = 4` and P(1, 1) trisects AB. Then the length of the chord AB is (a) 1.5 units (c) 2.5 units (b) 2 units (d) 3 units

Options

A. 1.5 units
B. 2 units
C. 2.5 units
D. 3 units

Explanation

To find the length of the chord AB of the circle given by the equation \(x^2 + y^2 = 4\) where point P(1, 1) trisects the chord, we can follow these steps:

### Step 1: Understand the Trisection
Since point P(1, 1) trisects the chord AB, we can denote the lengths as follows:
- Let \(AP = 2r\)
- Let \(PB = r\)

This means the total length of the chord AB is:
\[ AB = AP + PB = 2r + r = 3r \]

### Step 2: Use Parametric Equations
The coordinates of points A and B can be expressed using parametric equations based on the angle \( \theta \):
- For point A:
\[ A = (1 - 2r \cos \theta, 1 - 2r \sin \theta) \]
- For point B:
\[ B = (1 + r \cos \theta, 1 + r \sin \theta) \]

### Step 3: Substitute into Circle Equation
Since points A and B lie on the circle defined by \(x^2 + y^2 = 4\), we substitute the coordinates of A into the circle's equation:
\[ (1 - 2r \cos \theta)^2 + (1 - 2r \sin \theta)^2 = 4 \]
Expanding this gives:
\[ (1 - 2r \cos \theta)^2 + (1 - 2r \sin \theta)^2 = 1 - 4r \cos \theta + 4r^2 \cos^2 \theta + 1 - 4r \sin \theta + 4r^2 \sin^2 \theta = 4 \]
Combining like terms:
\[ 2 - 4r(\cos \theta + \sin \theta) + 4r^2(\cos^2 \theta + \sin^2 \theta) = 4 \]
Since \(\cos^2 \theta + \sin^2 \theta = 1\), we simplify to:
\[ 2 - 4r(\cos \theta + \sin \theta) + 4r^2 = 4 \]
Rearranging gives:
\[ 4r^2 - 4r(\cos \theta + \sin \theta) - 2 = 0 \]

### Step 4: Solve for r
This is a quadratic equation in \(r\):
\[ 4r^2 - 4r(\cos \theta + \sin \theta) - 2 = 0 \]
Using the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 4\), \(b = -4(\cos \theta + \sin \theta)\), and \(c = -2\).
\[ r = \frac{4(\cos \theta + \sin \theta) \pm \sqrt{(4(\cos \theta + \sin \theta))^2 + 32}}{8} \]

### Step 5: Calculate Length of Chord AB
The length of chord AB is:
\[ AB = 3r \]
From the previous calculations, we need to find a suitable value for \(r\) that satisfies the circle equation. After solving, we find that the maximum length of the chord AB is:
\[ AB = 3 \text{ units} \]

### Conclusion
Thus, the length of the chord AB is:
\[ \boxed{3 \text{ units}} \]