Q.Slope of a line which cuts off intercepts of equal lengths on the axes is
(A) $-1$
(B) $-0$
(C) $2$
(D) $\sqrt{3}$

The key idea is that intercepts are the distances where the line meets the $x$- and $y$-axes. If both intercepts are equal in length, say $a$, the line passes through $(a,0)$ and $(0,a)$ or through $(a,0)$ and $(0,-a)$, depending on sign.

Step 1: For equal positive intercepts $a$, points are $(a,0)$ and $(0,a)$.

Slope $m = \frac{a-0}{0-a} = \frac{a}{-a} = -1$.

Step 2: For intercepts $a$ and $-a$ (equal magnitude, opposite signs), points are $(a,0)$ and $(0,-a)$. …

The key idea here is simple: when a line cuts the x-axis and y-axis, the intercepts are the points where the line meets each axis. If the lengths of these intercepts are equal, the line forms an isosceles right triangle with the axes. The slope then depends on the sign of the intercepts.

Let’s walk through it step by step.

1. Write the intercept form of a line. The equation of a line with x-intercept $a$ and y-intercept $b$ is

Here, $a$ is the x-intercept (the point $(a,0)$) and $b$ is the y-intercept (the point $(0,b)$). Both $a$ and $b$ can be positive or negative, depending on which side of the origin the line cuts.

2. Apply the condition: equal lengths of intercepts. “Cuts off intercepts of equal lengths” means the absolute values of $a$ and $b$ are equal. So

This gives two possibilities:

• $b = a$ (both intercepts have the same sign)
• $b = -a$ (intercepts have opposite signs)

3. Find the slope in each case. The slope of a line in intercept form can be found by rewriting $\frac{x}{a} + \frac{y}{b} = 1$ in slope-intercept form $y = mx + c$. Multiply through by $ab$:

Solve for $y$:

So the slope $m = -\frac{b}{a}$.

• Case 1: $b = a$

  Then $m = -\frac{a}{a} = -1$.

• Case 2: $b = -a$

  Then $m = -\frac{(-a)}{a} = +1$.

4. Interpret the problem’s wording. The phrase “cuts off intercepts of equal lengths” in most exam contexts means the absolute lengths are equal, not necessarily that the signed intercepts are equal. So both $m = -1$ and $m = +1$ are mathematically valid. …