Question

The intercept cut off from $y$ axis is twice that from $x$ axis by the line and line is passing through $(1,2)$ then its equation is

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

Answer 1

To solve this problem, first we have to understand and learn the intercept form of the straight line and after that we have to find the value of intercepts of both the axis and then substitute the value of the given point in the intercept form equation and we will get our required answer.

Complete step-by-step solution:
A line can be defined as a one dimensional geometric shape. It is measured in respect to only one dimension (i.e. length). The points where the line crosses the two axes are called the intercepts. There are basically two intercepts, $x$-intercept and $y$-intercept. The point of intersection of the line with $X$-axis gives the $x$-intercept definition and the point of intersection of the line with $Y$-axis gives the $y$-intercept definition.
The point where the line or curve crosses the axis of the graph is called an intercept. If the axis is not specified, usually the $Y$axis is considered.
Let’s understand the intercept form of the straight line-
The intercept form of a straight line can be written as:
$\dfrac{x}{a}+\dfrac{y}{b}=1$
Here, $x$ and $y$ are coordinate axes, $a$ and $b$ are intercepted on the $x$and $y$ axis respectively.
The intercept of a line can also be found by substituting either ordinate or abscissa as zero in the equation for the line. If $y$ intercept is to be found, then the value of $x$ coordinate is substituted as zero and if the $x$ intercept is to be found, then the value of $y$ coordinate is substituted as zero in the general equation.
Now, according to the question:
Let the intercept made on $x$axis be $k$
And the intercept made on $y$ axis be $2k$
We know that if a line has $x$ and $y$ intercepts as $a$ and $b$ then the equation of the line is given as:
$\dfrac{x}{a}+\dfrac{y}{b}=1$
Here, $a=k$ and $b=2k$
Then now it becomes as:
$\Rightarrow \dfrac{x}{k}+\dfrac{y}{2k}=1$ $......................(i)$
Given that the line passes through the point $(1,2)$ . Then its equation is given by substituting $(1,2)$ in the equation $(i)$
We will get:
$\Rightarrow \dfrac{1}{k}+\dfrac{2}{2k}=1$
$\Rightarrow 2\times 1+2=2k$
$\Rightarrow 4=2k$
$\Rightarrow k=2$
Now, substituting $k=2$ in the equation $(i)$
We will get:
$\Rightarrow \dfrac{x}{k}+\dfrac{y}{2k}=1$
$\Rightarrow \dfrac{x}{2}+\dfrac{y}{4}=1$
$\Rightarrow 2x+y=4$
Therefore, equation of line is given as:
$\Rightarrow 2x+y=4$
Hence, the correct option is $1$.

Note: The slope intercept is the most “popular” form of a straight line. Many students find this useful because of its simplicity, because one can easily describe the characteristics of the straight line even without seeing the graph because with the help of this form, slope and intercept can easily be identified.