Question

The equation of the lines through \( (1, 1) \) and making angles of \( 45^\circ \) with the line \( x + y = 0 \) are:

• A. \( x-1=0, x-y=0 \)
• B. \( x-y=0, y-1=0 \)
• C. \( x+y-2=0, y-1=0 \)
• D. \( x-1=0, y-1=0 \)

Hint:

The line \( x+y=0 \) has an inclination of \( 135^\circ \) with the positive x-axis. Lines making an angle of \( 45^\circ \) with it must have inclinations:
\( 135^\circ - 45^\circ = 90^\circ \) (vertical line)
\( 135^\circ + 45^\circ = 180^\circ \) (horizontal line)
The vertical and horizontal lines passing through \( (1, 1) \) are simply \( x=1 \) and \( y=1 \).

Answer 1

The Correct Option is D

Step 1: Find the slope of the given line.
The line $x+y=0$ rewrites as $y=-x$, so its slope is $m_1=-1$, meaning it makes $135^\circ$ with the x-axis.
Step 2: Use the angle-of-inclination idea.
The required lines make $45^\circ$ with a line already at $135^\circ$, so their inclinations are $135^\circ + 45^\circ = 180^\circ$ or $135^\circ - 45^\circ = 90^\circ$.
Step 3: Translate those inclinations into slopes.
An inclination of $180^\circ$ means a horizontal line (slope $0$), and $90^\circ$ means a vertical line (undefined slope).
Step 4: Build the horizontal line through $(1,1)$.
A horizontal line through $(1,1)$ is $y = 1$, that is $y-1=0$.
Step 5: Build the vertical line through $(1,1)$.
A vertical line through $(1,1)$ is $x = 1$, that is $x-1=0$.
Step 6: Verify the $45^\circ$ condition.
Both $x-1=0$ and $y-1=0$ make $45^\circ$ with the line at $135^\circ$, confirming the pair. \[ \boxed{x-1=0\ \text{and}\ y-1=0} \]