The length of the chord joining the points ( `4cos theta , 4 sin theta ) ` and `[ 4 cos ( theta + 60^(@)), 4 sin ( theta + 60^(@))]` of the circle `x^(2) +y^(2) =16` is `:`

Options

A. 4
B. 6
C. 2
D. 8

Answer

To find the length of the chord joining the points \((4 \cos \theta, 4 \sin \theta)\) and \((4 \cos(\theta + 60^\circ), 4 \sin(\theta + 60^\circ))\) on the circle defined by the equation \(x^2 + y^2 = 16\), we will follow these steps:

### Step 1: Identify the Points
The points are given as:
- Point 1: \(P_1 = (4 \cos \theta, 4 \sin \theta)\)
- Point 2: \(P_2 = (4 \cos(\theta + 60^\circ), 4 \sin(\theta + 60^\circ))\)

### Step 2: Use the Angle Addition Formulas
We can express \(P_2\) using the angle addition formulas:
\[ \cos(\theta + 60^\circ) = \cos \theta \cos 60^\circ - \sin \theta \sin 60^\circ \]
\[ \sin(\theta + 60^\circ) = \sin \theta \cos 60^\circ + \cos \theta \sin 60^\circ \]

Substituting \(\cos 60^\circ = \frac{1}{2}\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\):
\[ P_2 = \left(4 \left(\cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2}\right), 4 \left(\sin \theta \cdot \frac{1}{2} + \cos \theta \cdot \frac{\sqrt{3}}{2}\right)\right) \]
This simplifies to:
\[ P_2 = \left(2 \cos \theta - 2\sqrt{3} \sin \theta, 2 \sin \theta + 2\sqrt{3} \cos \theta\right) \]

### Step 3: Calculate the Length of the Chord
The length of the chord can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting \(P_1\) and \(P_2\):
\[ \text{Length} = \sqrt{(2 \cos \theta - 2\sqrt{3} \sin \theta - 4 \cos \theta)^2 + (2 \sin \theta + 2\sqrt{3} \cos \theta - 4 \sin \theta)^2} \]
This simplifies to:
\[ = \sqrt{(-2 \cos \theta - 2\sqrt{3} \sin \theta)^2 + (-2 \sin \theta + 2\sqrt{3} \cos \theta)^2} \]

### Step 4: Expand and Simplify
Expanding the squares:
\[ = \sqrt{(2(\sqrt{3} \sin \theta + \cos \theta))^2 + (2(\sqrt{3} \cos \theta - \sin \theta))^2} \]
\[ = 2 \sqrt{(\sqrt{3} \sin \theta + \cos \theta)^2 + (\sqrt{3} \cos \theta - \sin \theta)^2} \]

### Step 5: Further Simplification
Calculating the squares:
\[ (\sqrt{3} \sin \theta + \cos \theta)^2 = 3 \sin^2 \theta + 2\sqrt{3} \sin \theta \cos \theta + \cos^2 \theta \]
\[ (\sqrt{3} \cos \theta - \sin \theta)^2 = 3 \cos^2 \theta - 2\sqrt{3} \sin \theta \cos \theta + \sin^2 \theta \]
Combining these:
\[ = 4 (\sin^2 \theta + \cos^2 \theta) + 2\sqrt{3} \sin \theta \cos \theta \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = 4 + 2\sqrt{3} \sin \theta \cos \theta \]

### Step 6: Final Calculation
Thus, the length of the chord is:
\[ = 2 \sqrt{4 + 2\sqrt{3} \sin \theta \cos \theta} \]
However, since the angle between the points is \(60^\circ\), we can directly calculate the length of the chord using the formula for the length of a chord in a circle:
\[ L = 2r \sin\left(\frac{\theta}{2}\right) \]
where \(r = 4\) and \(\theta = 60^\circ\):
\[ L = 2 \cdot 4 \cdot \sin(30^\circ) = 8 \cdot \frac{1}{2} = 4 \]

### Conclusion
The length of the chord is \(4\).