The line which cuts off equal intercept from the axes and pass through the point (1,-2) is __.

NCERT Exemplar · Q43

(1,-2)

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Concept understanding — Coordinate Axes Equations

Coordinate Axes Equations

Imagine a blank sheet of paper. You want to describe exactly where a point is. The most natural way is to draw two perpendicular lines — one horizontal, one vertical — and measure distances from them. Those two lines are the coordinate axes.

The horizontal line is the x-axis. The vertical line is the y-axis. Where they cross is the origin, the point $(0,0)$ .

Now, here's the key idea: every point on the x-axis has a y-coordinate of 0. Why? Because to be on that horizontal line, you haven't moved up or down at all — your vertical distance from the axis is zero. Similarly, every point on the y-axis has an x-coordinate of 0 — you haven't moved left or right from the vertical line.

That's the entire intuition. Let's make it precise.

The Equations

\text{x-axis: } y = 0

That's it. These are the equations of the coordinate axes.

$y = 0$ means: "all points where the y-coordinate is zero, regardless of x." That's the entire horizontal line through the origin.
$x = 0$ means: "all points where the x-coordinate is zero, regardless of y." That's the entire vertical line through the origin.

Watch out

A common mistake is to think $y=0$ means "the point $(0,0)$ ". No — it means every point with $y=0$ , like $(5,0)$ , $(-3,0)$ , $(100,0)$ , etc. It's a whole line, not a single point.

Why This Matters

These two equations are the foundation for everything in coordinate geometry. Every other line, curve, or shape is described relative to these axes. For example:

The line $y = 2$ is a horizontal line parallel to the x-axis, shifted up by 2 units.
The line $x = -3$ is a vertical line parallel to the y-axis, shifted left by 3 units.

So when you see $y = 0$ or $x = 0$ , think: "the original, unshifted axes themselves." …

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A line cutting equal intercepts from both axes has the form $x + y = a$ ; substituting the given point yields the equation $x + y + 1 = 0$ .

When a line cuts off equal intercepts from the coordinate axes, it means the $x$ -intercept and $y$ -intercept have the same absolute value. If we denote both intercepts as $a$ (they could be equal or opposite in sign), the intercept form of a line is:

\frac{x}{a} + \frac{y}{b} = 1

For equal intercepts, we have two cases to consider: either both intercepts are $a$ (same sign), or they are $a$ and $-a$ (opposite signs).

Case 1: Both intercepts equal $a$

The intercept form becomes:

\frac{x}{a} + \frac{y}{a} = 1

Simplifying:

x + y = a

This represents a family of lines with slope $-1$ .

Case 2: Intercepts are $a$ and $-a$

The intercept form becomes:

\frac{x}{a} + \frac{y}{-a} = 1

Simplifying:

x - y = a

This represents a family of lines with slope $1$ .

Now we determine which case applies by using the condition that the line passes through $(1, -2)$ .

Testing Case 1: Substitute $(1, -2)$ into $x + y = a$ :

1 + (-2) = a

a = -1

So the line is $x + y = -1$ , or equivalently $x + y + 1 = 0$ .

Testing Case 2: Substitute $(1, -2)$ into $x - y = a$ :

1 - (-2) = a

a = 3

So the line would be $x - y = 3$ , or $x - y - 3 = 0$ .

Verifying the intercepts:

For $x + y + 1 = 0$ : …

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Q.The line which cuts off equal intercept from the axes and pass through the point (1,−2)(1,-2)(1,−2) is ____.

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Q.The line which cuts off equal intercept from the axes and pass through the point $(1,-2)$ is ____.

Unlock everything free for 14 days