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 Question:
We must find the point of intersection of a line with the X-axis to establish the center of rotation. Then, we find the line's original angle of inclination, add the rotation angle, and formulate the equation of the new rotated line.

Step 2: Detailed Explanation:
1. Find the coordinates of M:
The line is $x - y - 2 = 0$.
It cuts the X-axis where $y = 0$.
Substitute $y = 0$: $x - 0 - 2 = 0 \Rightarrow x = 2$.
So, the pivot point is $M(2, 0)$.
2. Find the original angle of inclination ($\theta_1$):
Rewrite the line equation in slope-intercept form ($y = mx + c$):
$y = x - 2$.
The slope $m = 1$.
Since $m = \tan(\theta_1)$, we have $\tan(\theta_1) = 1 \Rightarrow \theta_1 = 45^{\circ}$.
3. Apply the rotation:
The line is rotated anticlockwise (positive direction) about M by $45^{\circ}$.
The new angle of inclination ($\theta_2$) is:
$\theta_2 = \theta_1 + 45^{\circ}$
$\theta_2 = 45^{\circ} + 45^{\circ} = 90^{\circ}$.
4. Form the new line equation:
A line with an inclination of $90^{\circ}$ is a perfectly vertical line parallel to the Y-axis.
The slope $m_{\text{new}} = \tan(90^{\circ})$ is undefined.
The general equation for a vertical line passing through any point $(x_1, y_1)$ is simply $x = x_1$.
Since our new line must pass through the pivot point $M(2, 0)$, its equation is strictly:
$x = 2$.

Step 3: Final Answer:
The equation of the line in the new position is $x = 2$, matching option (d).