A point P moves such that the chord of contact of P with respect to the circle `x^(2)+y^(2)=4` passes through the point (1, 1). The coordinates of P when it is nearest to the origin are

Options

A. (1, 2)
B. (2, 2)
C. (3, 3)
D. `(sqrt2, sqrt2)`

Answer

To solve the problem, we need to find the coordinates of point P such that the chord of contact from point P to the circle \(x^2 + y^2 = 4\) passes through the point (1, 1), and we want to determine the position of P when it is nearest to the origin.

### Step 1: Write the equation of the chord of contact
The equation of the chord of contact from a point \(P(x_1, y_1)\) to the circle \(x^2 + y^2 = r^2\) (where \(r^2 = 4\) in this case) is given by:
\[ xx_1 + yy_1 = r^2 \]
Substituting \(r^2 = 4\), we have:
\[ xx_1 + yy_1 = 4 \]

### Step 2: Substitute the point (1, 1)
Since the chord of contact passes through the point (1, 1), we substitute \(x = 1\) and \(y = 1\) into the equation:
\[ 1 \cdot x_1 + 1 \cdot y_1 = 4 \]
This simplifies to:
\[ x_1 + y_1 = 4 \]

### Step 3: Express \(y_1\) in terms of \(x_1\)
From the equation \(x_1 + y_1 = 4\), we can express \(y_1\) as:
\[ y_1 = 4 - x_1 \]

### Step 4: Find the distance from the origin
The distance \(OP\) from the origin to the point \(P(x_1, y_1)\) is given by:
\[ OP = \sqrt{x_1^2 + y_1^2} \]
Substituting \(y_1 = 4 - x_1\):
\[ OP = \sqrt{x_1^2 + (4 - x_1)^2} \]
Expanding this:
\[ OP = \sqrt{x_1^2 + (16 - 8x_1 + x_1^2)} = \sqrt{2x_1^2 - 8x_1 + 16} \]

### Step 5: Minimize the distance
To minimize \(OP\), we can minimize the square of the distance:
\[ D = 2x_1^2 - 8x_1 + 16 \]
Taking the derivative and setting it to zero:
\[ \frac{dD}{dx_1} = 4x_1 - 8 = 0 \]
Solving for \(x_1\):
\[ 4x_1 = 8 \implies x_1 = 2 \]

### Step 6: Find \(y_1\)
Now substituting \(x_1 = 2\) back into the equation for \(y_1\):
\[ y_1 = 4 - x_1 = 4 - 2 = 2 \]

### Conclusion
Thus, the coordinates of point \(P\) when it is nearest to the origin are:
\[ \boxed{(2, 2)} \]