Question

The locus of a point that is equidistant from the lines \[ x + y - 2\sqrt{2} = 0 \quad {and} \quad x + y - \sqrt{2} = 0 { is:} \]

• A. \( x + y - 5\sqrt{2} = 0 \)
• B. \( x + y - 3\sqrt{2} = 0 \)
• C. \( 2x + 2y - 3\sqrt{2} = 0 \)
• D. \( 2x + 2y - 5\sqrt{2} = 0 \)

Hint:

For the locus of a point equidistant from two lines, find the average of the constants in the equations of the lines.

Answer 1

The Correct Option is C

Step 1: The locus of points equidistant from two parallel lines is the midline, which is the average of the equations of the two lines. The equations of the lines are: \[ x + y - 2\sqrt{2} = 0 \quad {and} \quad x + y - \sqrt{2} = 0. \] Step 2: To find the midline, take the average of the constants: \[ \frac{2\sqrt{2} + \sqrt{2}}{2} = \frac{3\sqrt{2}}{2}. \] Thus, the equation of the locus is: \[ x + y - 3\sqrt{2} = 0. \]