Question

The line MN whose equation is $x - y - 2 = 0$ cuts the X-axis at M and coordinates of N are (4, 2). The line MN is rotated about M through $45^{\circ}$ in anticlockwise direction. The equation of the line MN in the new position is ______.

• A. $y = -\sqrt{2}$
• B. $y = 2$
• C. $x = -2$
• D. $x = 2$

Hint:

Drawing a quick sketch makes this obvious! The original line passes through $(2,0)$ and $(0,-2)$. Rotating it upwards by $45^{\circ}$ around its X-intercept "stands it up" perfectly straight into a vertical line!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
First find the coordinates of $M$ and the initial slope. Then adjust the angle of inclination for the new line.

Step 2: Formula Application:
Line $x - y - 2 = 0$. At X-axis, $y=0 \implies x=2$. So $M = (2, 0)$. Initial slope $m = 1 \implies \theta = 45^\circ$.

Step 3: Explanation:
The line is rotated $45^\circ$ anticlockwise. New angle $\theta' = 45^\circ + 45^\circ = 90^\circ$. A line with an inclination of $90^\circ$ is a vertical line. Since it passes through $M(2, 0)$, its equation is $x = 2$.

Step 4: Final Answer:
The new equation is $x = 2$.