Question

The equations of the lines which cut off an intercept 1 from the y-axis and are equally inclined to the axes are:

• A. \( x - y + 1 = 0, x + y + 1 = 0 \)
• B. \( x - y - 1 = 0, x + y - 1 = 0 \)
• C. \( x - y - 1 = 0, x + y + 1 = 0 \)
• D. None of these

Hint:

For lines equally inclined to the axes, the slope will be \( \pm 1 \), which results in equations of the form \( y = x + c \) or \( y = -x + c \).

Answer 1

The Correct Option is C

Step 1: The general equation of a line is given by: \[ y = mx + c, \] where \( m \) is the slope and \( c \) is the y-intercept. Since the line cuts an intercept of 1 on the y-axis, we have \( c = 1 \). Thus, the equation of the line becomes: \[ y = mx + 1. \] Step 2: The lines are equally inclined to the axes, meaning the angle between the line and the x-axis is the same as the angle between the line and the y-axis. This occurs when the slope \( m \) is \( \pm 1 \), because the tangent of \( 45^\circ \) is 1. So, the equations of the lines are: \[ y = x + 1 \quad {and} \quad y = -x + 1. \] Step 3: Rewriting these equations in general form: \[ x - y + 1 = 0 \quad {and} \quad x + y + 1 = 0. \] Thus, the correct answer is \( x - y - 1 = 0, x + y + 1 = 0 \).