Finding the Equation of a Straight Line

The question asks for the equation of a straight line that passes through a specific point and has a particular property regarding its intercepts on the coordinate axes. The point given is (1, -2), and the property is that it cuts off equal intercepts from the axes.

Understanding Equal Intercepts

A straight line intersects the x-axis at some point and the y-axis at some point. These points define the intercepts. The x-intercept is the x-coordinate of the point where the line crosses the x-axis (where y=0). The y-intercept is the y-coordinate of the point where the line crosses the y-axis (where x=0).

When a line cuts off equal intercepts from the axes, it means the distance from the origin to the x-intercept is the same as the distance from the origin to the y-intercept. Let's denote this equal intercept value as 'a'.

• The x-intercept is 'a'. This means the line passes through the point (a, 0).
• The y-intercept is 'a'. This means the line passes through the point (0, a).

Note: The intercepts are signed distances from the origin. If the intercepts are equal, they have the same value, 'a'.

Using the Intercept Form of a Line

The equation of a straight line with x-intercept 'a' (where $a \neq 0$) and y-intercept 'b' (where $b \neq 0$) is given by the intercept form:

\begin{equation*}\frac{x}{a} + \frac{y}{b} = 1\end{equation*}

In this problem, the line cuts off equal intercepts, so the x-intercept is equal to the y-intercept. Let's call this value 'a'. So, we have $a=b$. Substituting this into the intercept form equation, we get:

\begin{equation*}\frac{x}{a} + \frac{y}{a} = 1\end{equation*}

Assuming 'a' is not zero (a line cutting zero intercepts and passing through (1, -2) would pass through the origin, which (1, -2) is not), we can multiply the entire equation by 'a' to simplify it:

\begin{equation*}a \left( \frac{x}{a} + \frac{y}{a} \right) = a \times 1\end{equation*}

\begin{equation*}x + y = a\end{equation*}

This is the general form of a line that cuts off equal intercepts 'a' from the axes.

Using the Given Point to Find 'a'

We know that the straight line passes through the point (1, -2). This means the coordinates of this point must satisfy the equation of the line. We substitute $x=1$ and $y=-2$ into the equation $x + y = a$:

\begin{equation*}1 + (-2) = a\end{equation*}

\begin{equation*}1 - 2 = a\end{equation*}

\begin{equation*}-1 = a\end{equation*}

So, the value of the equal intercept 'a' is -1.

Writing the Final Equation

Now that we have the value of 'a', which is -1, we substitute it back into the simplified equation of the line $x + y = a$:

\begin{equation*}x + y = -1\end{equation*}

To match the format of the given options, we can move the constant term to the left side of the equation:

\begin{equation*}x + y + 1 = 0\end{equation*}

This is the equation of the straight line that passes through the point (1, -2) and cuts off equal intercepts from the axes.

Checking the Options

Let's compare our derived equation, $x + y + 1 = 0$, with the given options:

1. x + y - 1 = 0
2. x - y - 1 = 0
3. x + y + 1 = 0
4. x - y - 2 = 0

Our derived equation matches option 3.

Revision Table: Straight Line Equation Concepts

Additional Information on Intercepts

The intercepts of a line are important features that help define its position on the coordinate plane. The x-intercept is where the line crosses the x-axis, meaning the y-coordinate is 0. The y-intercept is where the line crosses the y-axis, meaning the x-coordinate is 0.

• To find the x-intercept from the equation $Ax + By + C = 0$, set $y=0$ and solve for $x$. The x-intercept is $-C/A$ (if $A \neq 0$).
• To find the y-intercept from the equation $Ax + By + C = 0$, set $x=0$ and solve for $y$. The y-intercept is $-C/B$ (if $B \neq 0$).

In our derived equation $x + y + 1 = 0$, we have $A=1$, $B=1$, $C=1$.

• x-intercept: Set $y=0 \implies x + 0 + 1 = 0 \implies x = -1$. The x-intercept is -1.
• y-intercept: Set $x=0 \implies 0 + y + 1 = 0 \implies y = -1$. The y-intercept is -1.

The x-intercept and y-intercept are indeed equal, both being -1. This confirms our solution is correct based on the problem statement about equal intercepts.