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 \( P(h, k) \) which moves such that the chord of contact with respect to the circle \( x^2 + y^2 = 4 \) passes through the point \( (1, 1) \). We also want to find the coordinates of \( P \) when it is nearest to the origin.

### Step-by-step Solution:

1. **Equation of the Circle**:
The given circle is \( x^2 + y^2 = 4 \). This can be rewritten as \( x^2 + y^2 = r^2 \) where \( r = 2 \).

2. **Chord of Contact**:
The chord of contact from an external point \( P(h, k) \) to the circle is given by the equation:
\[ hx + ky = r^2 \]
For our circle, this becomes:
\[ hx + ky = 4 \]

3. **Condition for the Chord of Contact**:
Since the chord of contact passes through the point \( (1, 1) \), we substitute \( x = 1 \) and \( y = 1 \) into the chord of contact equation:
\[ h(1) + k(1) = 4 \]
This simplifies to:
\[ h + k = 4 \]

4. **Expressing \( h \) in terms of \( k \)**:
From the equation \( h + k = 4 \), we can express \( h \) as:
\[ h = 4 - k \]

5. **Distance from the Origin**:
The distance \( d \) from the origin \( (0, 0) \) to the point \( P(h, k) \) is given by:
\[ d = \sqrt{h^2 + k^2} \]
Substituting \( h = 4 - k \):
\[ d = \sqrt{(4 - k)^2 + k^2} \]

6. **Expanding the Distance**:
We expand the expression:
\[ d = \sqrt{(16 - 8k + k^2) + k^2} = \sqrt{16 - 8k + 2k^2} \]

7. **Minimizing the Distance**:
To find the minimum distance, we can minimize the expression under the square root:
\[ f(k) = 2k^2 - 8k + 16 \]
This is a quadratic function in \( k \).

8. **Finding the Vertex**:
The vertex of a quadratic \( ax^2 + bx + c \) occurs at \( k = -\frac{b}{2a} \):
\[ k = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2 \]

9. **Finding \( h \)**:
Substitute \( k = 2 \) back into the equation for \( h \):
\[ h = 4 - k = 4 - 2 = 2 \]

10. **Coordinates of Point \( P \)**:
Thus, the coordinates of point \( P \) when it is nearest to the origin are:
\[ (h, k) = (2, 2) \]

### Final Answer:
The coordinates of \( P \) when it is nearest to the origin are \( (2, 2) \).