Find the equation of the straight line which passes through the point (1,-2) and cuts off equal intercepts fro

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(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."

Tip

To remember: $y=0$ is horizontal (the x-axis), $x=0$ is vertical (the y-axis). The variable that equals zero tells you which axis you're on — the other variable is free to roam.

A Quick Check

Which axis does the point $(0, 7)$ lie on?

It has $x=0$ , so it's on the y-axis.

Which axis does $(-4, 0)$ lie on?

It has $y=0$ , so it's on the x-axis.

That's the whole concept. Two simple equations, but they define the entire coordinate system.

The Equations of the Coordinate Axes are introduced right at the start of the NCERT Class 11 Mathematics chapter on Straight Lines, matching searches like "equation of x-axis and y-axis" or "coordinate geometry basics class 11 maths". Though simple, this idea is a common short-answer topic in CBSE boards and forms the groundwork for every later coordinate geometry and conic section problem in JEE Main.

A line cutting equal intercepts from both axes has slope $\pm 1$ ; passing through $(1, -2)$ gives two lines: $x + y + 1 = 0$ and $x - y - 3 = 0$ .

Understanding Equal Intercepts

When a line cuts off equal intercepts from the coordinate axes, it means the $x$ -intercept and $y$ -intercept have the same absolute value. There are two distinct cases to consider:

Case 1: The intercepts are equal and have the same sign (both $a$ or both $-a$ ).

Case 2: The intercepts are equal in magnitude but opposite in sign.

The intercept form of a line is:

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

where $a$ is the $x$ -intercept and $b$ is the $y$ -intercept.

For equal intercepts, we need $a = b$ or $a = -b$ .

Solution

1. When both intercepts are equal: $a = b$

The intercept form becomes:

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

Simplifying:

x + y = a

Since this line must pass through $(1, -2)$ , substitute these coordinates:

1 + (-2) = a

a = -1

Therefore, the first line is:

x + y = -1

or equivalently:

x + y + 1 = 0

2. When intercepts are equal in magnitude but opposite in sign: $a = -b$

The intercept form becomes:

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

Simplifying:

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

x - y = a

Since this line must pass through $(1, -2)$ , substitute:

1 - (-2) = a

a = 3

Therefore, the second line is:

x - y = 3

or equivalently:

x - y - 3 = 0

Tip

Lines with equal intercepts always have slope $\pm 1$ . You can verify: $x + y = c$ has slope $-1$ , while $x - y = c$ has slope $+1$ .

Watch out

Don't forget the second case! Many students only consider $a = b$ and miss the line where intercepts are equal but opposite in sign.

Verification:

For $x + y + 1 = 0$ : Setting $y = 0$ gives $x = -1$ ; setting $x = 0$ gives $y = -1$ . Both intercepts are $-1$ . ✓

For $x - y - 3 = 0$ : Setting $y = 0$ gives $x = 3$ ; setting $x = 0$ gives $y = -3$ . Intercepts are $3$ and $-3$ (equal magnitude). ✓

✓Final answer

\boxed{x + y + 1 = 0}

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Q.Find the equation of the straight line which passes through the point $(1,-2)$ and cuts off equal intercepts from axes.

Concept understanding — Coordinate Axes Equations

Coordinate Axes Equations

The Equations

Why This Matters

A Quick Check

Equal Intercepts from Axes

Solution

Understanding Equal Intercepts

Solution

Q.Find the equation of the straight line which passes through the point (1,−2)(1,-2)(1,−2) and cuts off equal intercepts from axes.

Understanding Equal Intercepts

Solution

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Q.Find the equation of the straight line which passes through the point $(1,-2)$ and cuts off equal intercepts from axes.

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