Key Concepts and Formulas

• Distance from a Point to a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
• Relationship between Chord Length and Distance from Center: If a chord of length $L$ is at a distance $d$ from the center of a circle with radius $R$, then $R^2 = d^2 + (\frac{L}{2})^2$.
• Distance Between Parallel Lines: The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is given by $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Step 1: Find the distance of chord PQ from the origin.

The equation of line PQ is $x + y = 2$, or $x + y - 2 = 0$. The center of the circle is at the origin (0, 0). Using the point-to-line distance formula, the distance $d_{PQ}$ of the chord PQ from the origin is:

$d_{PQ} = \frac{|1(0) + 1(0) - 2|}{\sqrt{1^2 + 1^2}} = \frac{|-2|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$

Step 2: Find the distance of chord MN from the origin.

The chord MN has length 2. Let the distance of the chord MN from the origin be $d_{MN}$. The radius of the circle is $\sqrt{10}$. Using the relationship between chord length and distance from the center:

$(\sqrt{10})^2 = d_{MN}^2 + (\frac{2}{2})^2$ $10 = d_{MN}^2 + 1$ $d_{MN}^2 = 9$ $d_{MN} = 3$

Step 3: Determine the equations of the lines representing MN.

Since the slope of MN is -1, the equation of the line MN can be written in the form $x + y + C = 0$. We know the distance from the origin to this line is 3. Using the point-to-line distance formula:

$3 = \frac{|1(0) + 1(0) + C|}{\sqrt{1^2 + 1^2}} = \frac{|C|}{\sqrt{2}}$ $|C| = 3\sqrt{2}$ So, $C = 3\sqrt{2}$ or $C = -3\sqrt{2}$. Thus, the two possible equations for the line MN are $x + y + 3\sqrt{2} = 0$ and $x + y - 3\sqrt{2} = 0$.

Step 4: Calculate the distance between the parallel lines.

The equation of line PQ is $x + y - 2 = 0$. The equations of the two possible lines for MN are $x + y + 3\sqrt{2} = 0$ and $x + y - 3\sqrt{2} = 0$. We need to find the distance between PQ and either of the lines representing MN.

• Distance between $x + y - 2 = 0$ and $x + y + 3\sqrt{2} = 0$: $d_1 = \frac{|-2 - 3\sqrt{2}|}{\sqrt{1^2 + 1^2}} = \frac{|-2 - 3\sqrt{2}|}{\sqrt{2}} = \frac{2 + 3\sqrt{2}}{\sqrt{2}} = \sqrt{2} + 3$

• Distance between $x + y - 2 = 0$ and $x + y - 3\sqrt{2} = 0$: $d_2 = \frac{|-2 - (-3\sqrt{2})|}{\sqrt{1^2 + 1^2}} = \frac{|-2 + 3\sqrt{2}|}{\sqrt{2}} = \frac{3\sqrt{2} - 2}{\sqrt{2}} = 3 - \sqrt{2}$

Since the options are $3-\sqrt{2}$, $2-\sqrt{3}$, $\sqrt{2}-1$, and $\sqrt{2}+1$, the correct distance must be $3 - \sqrt{2}$.

Common Mistakes & Tips

• Sign Errors: Be careful with signs when using the point-to-line distance formula and the distance between parallel lines formula.
• Incorrect Formula: Make sure to use the correct formulas for distance from a point to a line and distance between parallel lines.
• Two Possible Lines: Remember that there are two possible lines MN, one on each side of the origin, equidistant from it.

Summary

We first calculated the distance of chord PQ from the origin using the point-to-line distance formula. Then, we found the distance of chord MN from the origin using the relationship between chord length and distance from the center. We then found the equations of the two lines representing MN. Finally, we calculated the distance between PQ and each of the lines representing MN and found that the correct distance is $3 - \sqrt{2}$.

Final Answer

The final answer is $\boxed{3-\sqrt{2}}$, which corresponds to option (A).