The angle at which the circle `x^(2)+y^(2)` = 16 can be seen from the point (8, 0) is

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(pi)/(6)`

Answer

To find the angle at which the circle \(x^2 + y^2 = 16\) can be seen from the point \((8, 0)\), we can follow these steps:

### Step 1: Understand the Circle and the Point
The equation \(x^2 + y^2 = 16\) represents a circle centered at the origin \((0, 0)\) with a radius of \(4\) (since \(\sqrt{16} = 4\)). The point \((8, 0)\) lies on the x-axis, to the right of the circle.

### Step 2: Differentiate the Circle's Equation
To find the slope of the tangent to the circle at any point, we differentiate the equation of the circle with respect to \(x\):
\[ \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(16) \]
This gives:
\[ 2x + 2y \frac{dy}{dx} = 0 \]
Rearranging this, we find:
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Step 3: Find Points of Tangency
To find the angle subtended by the circle at the point \((8, 0)\), we need to determine the points where the tangents from \((8, 0)\) touch the circle. The distance from the point \((8, 0)\) to the center of the circle \((0, 0)\) is \(8\).

### Step 4: Use the Power of a Point Theorem
The power of the point theorem states that the power of a point \(P\) with respect to a circle is given by:
\[ \text{Power} = OP^2 - r^2 \]
where \(O\) is the center of the circle and \(r\) is the radius. Here, \(OP = 8\) and \(r = 4\):
\[ \text{Power} = 8^2 - 4^2 = 64 - 16 = 48 \]
The length of the tangent from the point \((8, 0)\) to the circle can be calculated as:
\[ \text{Length of tangent} = \sqrt{\text{Power}} = \sqrt{48} = 4\sqrt{3} \]

### Step 5: Find the Angle
The angle \(\theta\) subtended by the tangents at the point \((8, 0)\) can be found using the formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\text{Length of tangent}}{\text{Distance from point to center}} \]
Substituting the values:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{4\sqrt{3}}{8} = \frac{\sqrt{3}}{2} \]
This implies:
\[ \frac{\theta}{2} = 60^\circ \quad \text{(since } \tan(60^\circ) = \sqrt{3}\text{)} \]
Thus, the angle \(\theta\) is:
\[ \theta = 120^\circ \]

### Step 6: Convert to Radians
To express the angle in radians:
\[ \theta = \frac{120 \times \pi}{180} = \frac{2\pi}{3} \]

### Final Answer
The angle at which the circle \(x^2 + y^2 = 16\) can be seen from the point \((8, 0)\) is \(\frac{2\pi}{3}\) radians.

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