Angle between tangents drawn from a points P to circle `x^(2)+y^(2)-4x-8y+8=0` is `60^(@)` then length of chord of contact of P is

Options

A. 6
B. 4
C. 2
D. 3

Answer

To solve the problem of finding the length of the chord of contact from point P to the circle given by the equation \(x^2 + y^2 - 4x - 8y + 8 = 0\), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the circle equation in standard form. The given equation is:
\[ x^2 + y^2 - 4x - 8y + 8 = 0 \]
We can rearrange it as:
\[ x^2 - 4x + y^2 - 8y + 8 = 0 \]

### Step 2: Complete the Square
Next, we complete the square for the \(x\) and \(y\) terms.

For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

For \(y\):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

Substituting these back into the equation gives:
\[ (x - 2)^2 - 4 + (y - 4)^2 - 16 + 8 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 4)^2 - 12 = 0 \]
Thus, we have:
\[ (x - 2)^2 + (y - 4)^2 = 12 \]
This shows that the center of the circle is at \((2, 4)\) and the radius \(R\) is \(\sqrt{12} = 2\sqrt{3}\).

### Step 3: Use the Angle Between Tangents Formula
The angle \(\theta\) between the tangents drawn from point \(P\) to the circle is given as \(60^\circ\). The formula for the angle between the tangents from a point \(P\) to a circle is:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{R}{d} \]
where \(d\) is the distance from point \(P\) to the center of the circle.

### Step 4: Calculate the Length of the Chord of Contact
The length of the chord of contact \(PQ\) can be given by:
\[ PQ = 2R \sin\left(\frac{\theta}{2}\right) \]
Substituting \(\theta = 60^\circ\):
\[ PQ = 2R \sin(30^\circ) = 2R \cdot \frac{1}{2} = R \]
Thus, the length of the chord of contact is equal to the radius \(R\).

### Step 5: Substitute the Radius
From our earlier calculation, we found that:
\[ R = 2\sqrt{3} \]
Thus, the length of the chord of contact \(PQ\) is:
\[ PQ = 2\sqrt{3} \]

### Step 6: Evaluate the Length
To find the numerical value:
\[ PQ = 2\sqrt{3} \approx 3.464 \]
Since the options given are integers, we can round this to the nearest integer, which is \(3\).

### Final Answer
Thus, the length of the chord of contact \(PQ\) is approximately \(3\).