Tangents `PA and PB` are drawn to the circle `x^2 +y^2=8` from any arbitrary point P on the line `x+y =4`. The locus of mid-point of chord of contact AB is

Options

A. `x^(2)+y^(2)+2x+2y=0`
B. `x^(2)+y^(2)-2x-2y=0`
C. `x^(2)+y^(2)-2x+2y=0`
D. `x^(2)+y^(2)+2x-2y=0`

Explanation

To solve the problem of finding the locus of the midpoint of the chord of contact AB from point P on the line \( x + y = 4 \) to the circle \( x^2 + y^2 = 8 \), we can follow these steps:

### Step 1: Identify the Circle and the Line
The equation of the circle is given by:
\[ x^2 + y^2 = 8 \]
The line from which the tangents are drawn is:
\[ x + y = 4 \]

### Step 2: Find the Equation of the Chord of Contact
The chord of contact from point \( P(x_1, y_1) \) to the circle can be expressed as:
\[ xx_1 + yy_1 = 8 \]
where \( (x_1, y_1) \) lies on the line \( x + y = 4 \).

### Step 3: Substitute for \( y_1 \)
Since \( P \) lies on the line \( x + y = 4 \), we can express \( y_1 \) in terms of \( x_1 \):
\[ y_1 = 4 - x_1 \]

### Step 4: Substitute \( y_1 \) in the Chord of Contact Equation
Substituting for \( y_1 \) in the chord of contact equation:
\[ xx_1 + y(4 - x_1) = 8 \]
This simplifies to:
\[ xx_1 + 4y - xy_1 = 8 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ xx_1 - xy_1 + 4y = 8 \]
This can be rewritten as:
\[ x(x_1 - y) + 4y = 8 \]

### Step 6: Finding the Midpoint of the Chord of Contact
Let the midpoint of the chord of contact be \( (h, k) \). The equation of the chord of contact can also be expressed as:
\[ hx + ky = h^2 + k^2 \]
Equating the two forms of the chord of contact gives:
\[ hx + ky = 8 \]

### Step 7: Equating Coefficients
From the equations:
1. \( hx + ky = 8 \)
2. \( xx_1 + yy_1 = 8 \)

We can equate coefficients:
\[ \frac{x_1}{h} = \frac{y_1}{k} = \frac{8}{h^2 + k^2} \]

### Step 8: Substitute \( y_1 \) Again
Substituting \( y_1 = 4 - x_1 \) into the equations gives:
\[ \frac{x_1}{h} = \frac{4 - x_1}{k} \]

### Step 9: Solve for \( x_1 \)
Cross-multiplying gives:
\[ kx_1 = h(4 - x_1) \]
Rearranging gives:
\[ kx_1 + hx_1 = 4h \implies x_1(k + h) = 4h \implies x_1 = \frac{4h}{h + k} \]

### Step 10: Substitute Back to Find \( k \)
Now substituting \( x_1 \) back into:
\[ \frac{4}{h + k} = \frac{8}{h^2 + k^2} \]
Cross-multiplying gives:
\[ 4(h^2 + k^2) = 8(h + k) \]
Simplifying gives:
\[ h^2 + k^2 - 2h - 2k = 0 \]

### Step 11: Final Locus Equation
Replacing \( h \) and \( k \) with \( x \) and \( y \) gives:
\[ x^2 + y^2 - 2x - 2y = 0 \]

### Conclusion
Thus, the locus of the midpoint of the chord of contact is:
\[ x^2 + y^2 - 2x - 2y = 0 \]