The length of the chord joining the points `(4 cos alpha, 4 sin alpha) and {4 cos (alpha+60^(@)), 4 sin (alpha+60^(@))}` on the circle `x^(2)+y^(2)=16` is

Options

A. 2
B. 3
C. 4
D. none

Answer

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

### Step 1: Identify the coordinates of the points
The coordinates of the points are given as:
- \( P = (4 \cos \alpha, 4 \sin \alpha) \)
- \( Q = (4 \cos(\alpha + 60^\circ), 4 \sin(\alpha + 60^\circ)) \)

### Step 2: Use the distance formula
The length of the chord \( PQ \) can be calculated using the distance formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( P \) and \( Q \):
\[ PQ = \sqrt{(4 \cos(\alpha + 60^\circ) - 4 \cos \alpha)^2 + (4 \sin(\alpha + 60^\circ) - 4 \sin \alpha)^2} \]

### Step 3: Factor out the common term
We can factor out \( 4 \) from both terms:
\[ PQ = 4 \sqrt{(\cos(\alpha + 60^\circ) - \cos \alpha)^2 + (\sin(\alpha + 60^\circ) - \sin \alpha)^2} \]

### Step 4: Use the angle addition formulas
Using the angle addition formulas:
- \( \cos(\alpha + 60^\circ) = \cos \alpha \cos 60^\circ - \sin \alpha \sin 60^\circ = \frac{1}{2} \cos \alpha - \frac{\sqrt{3}}{2} \sin \alpha \)
- \( \sin(\alpha + 60^\circ) = \sin \alpha \cos 60^\circ + \cos \alpha \sin 60^\circ = \frac{1}{2} \sin \alpha + \frac{\sqrt{3}}{2} \cos \alpha \)

### Step 5: Substitute the values into the distance formula
Now substituting these into our distance formula:
\[ PQ = 4 \sqrt{\left(\frac{1}{2} \cos \alpha - \frac{\sqrt{3}}{2} \sin \alpha - \cos \alpha\right)^2 + \left(\frac{1}{2} \sin \alpha + \frac{\sqrt{3}}{2} \cos \alpha - \sin \alpha\right)^2} \]
This simplifies to:
\[ PQ = 4 \sqrt{\left(-\frac{1}{2} \cos \alpha - \frac{\sqrt{3}}{2} \sin \alpha\right)^2 + \left(-\frac{1}{2} \sin \alpha + \frac{\sqrt{3}}{2} \cos \alpha\right)^2} \]

### Step 6: Simplify the expression
Calculating the squares:
1. For the first term:
\[ \left(-\frac{1}{2} \cos \alpha - \frac{\sqrt{3}}{2} \sin \alpha\right)^2 = \frac{1}{4} \cos^2 \alpha + \frac{3}{4} \sin^2 \alpha + \frac{\sqrt{3}}{2} \cos \alpha \sin \alpha \]
2. For the second term:
\[ \left(-\frac{1}{2} \sin \alpha + \frac{\sqrt{3}}{2} \cos \alpha\right)^2 = \frac{1}{4} \sin^2 \alpha + \frac{3}{4} \cos^2 \alpha - \frac{\sqrt{3}}{2} \sin \alpha \cos \alpha \]

### Step 7: Combine the terms
Combining these two results gives:
\[ PQ^2 = 4 \left(\frac{1}{4} (\cos^2 \alpha + \sin^2 \alpha) + \frac{1}{4} (\sin^2 \alpha + \cos^2 \alpha)\right) = 4 \cdot 1 = 16 \]
Thus, \( PQ = 4 \).

### Final Result
The length of the chord \( PQ \) is \( 4 \).