If `(pi)/6` and `(pi)/2` are the ends of chord of the circle `x^(2)+y^(2)=16` then its length is

Options

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

Answer

To find the length of the chord whose endpoints are given as \(\frac{\pi}{6}\) and \(\frac{\pi}{2}\) in the circle defined by the equation \(x^2 + y^2 = 16\), we can follow these steps:

### Step 1: Identify the radius of the circle
The equation of the circle is given as \(x^2 + y^2 = 16\). We can identify the radius \(r\) from this equation. The standard form of a circle is \(x^2 + y^2 = r^2\).

\[ r^2 = 16 \implies r = \sqrt{16} = 4 \]

### Step 2: Determine the angles
The endpoints of the chord are given as \(\theta_1 = \frac{\pi}{6}\) and \(\theta_2 = \frac{\pi}{2}\).

### Step 3: Calculate the difference between the angles
We need to find the difference \(\theta_1 - \theta_2\):

\[ \theta_1 - \theta_2 = \frac{\pi}{6} - \frac{\pi}{2} \]

To subtract these angles, we convert \(\frac{\pi}{2}\) to a fraction with a common denominator:

\[ \frac{\pi}{2} = \frac{3\pi}{6} \]

Now, we can perform the subtraction:

\[ \theta_1 - \theta_2 = \frac{\pi}{6} - \frac{3\pi}{6} = -\frac{2\pi}{6} = -\frac{\pi}{3} \]

Since we are interested in the absolute value for the sine function, we take:

\[ |\theta_1 - \theta_2| = \frac{\pi}{3} \]

### Step 4: Use the formula for the length of the chord
The formula for the length of the chord is given by:

\[ \text{Length of chord} = 2r \sin\left(\frac{\theta_1 - \theta_2}{2}\right) \]

Substituting the values we have:

\[ \text{Length of chord} = 2 \times 4 \times \sin\left(\frac{\frac{\pi}{3}}{2}\right) = 8 \sin\left(\frac{\pi}{6}\right) \]

### Step 5: Calculate \(\sin\left(\frac{\pi}{6}\right)\)
We know that:

\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 6: Substitute and calculate the final length
Now substitute this value back into the equation for the length of the chord:

\[ \text{Length of chord} = 8 \times \frac{1}{2} = 4 \]

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

---