Question

If \[x+2y=0\], then choose the correct option.
a) The line passes through \[(0,0)\] and \[m=-3\]
b) The line passes through \[(0,0)\] and \[m=\dfrac{-1}{2}\]
c) The line passes through \[(0,0)\] and \[m=3\]
d) None of these

Answer 1

Hint: We are given an equation of a line through which we are able to calculate its slope and will get to know whether it passes through the origin or not. First substitute the coordinates of the origin that is \[(0,0)\] if this satisfies the equation means on putting this in place of \[x\] and \[y\] respectively if \[LHS\] and \[RHS\] are equal then the particular point lies on the line. And to calculate the slope of the line whose equation is given just compare it with the standard equation of line and that is \[y=mx+c\] in this equation \[m\] is the slope of the equation. On comparing we will get our answer.

Complete step by step solution:
Given that, the equation of line is \[x+2y=0\]
Now to check whether it passes through the origin or not, substitute the coordinates of the origin
Since origin has coordinates \[(0,0)\]
Now putting these,
\[\Rightarrow 0+2(0)=0\]
\[\Rightarrow 0=0\]
Since the LHS and RHS are equal, that means these coordinates satisfies the equation and,
 hence the equation passes through the origin
Now to calculate the slope \[(m)\] of the equation
Since we know that the standard form of the equation of line is \[y=mx+c\] , where \[m\] is the slope of equation and \[c\] is the \[y\] intercept at which the line cuts the\[y\] axis.
Now, \[x+2y=0\] this can be written as
\[\Rightarrow 2y=-x\]
\[\Rightarrow y=-\dfrac{x}{2}\]
This expression and can be written as
\[\Rightarrow y=-\dfrac{1}{2}x\]
Now comparing this with its standard form \[y=mx+c\]
We get,
\[m=-\dfrac{1}{2}\]
\[c=0\]

Here also \[c=0\] that means the line passes through origin.
Hence option (b) is correct.

Note:
When we have to solve this type of question, just remember the standard form of the equation of a line and we can find it by comparing. To find the slope of an equation \[ax+by+c=0\]
Remember that slope is the ratio of minus of coefficient of \[x\] to coefficient of \[y\] that is \[-{}^{a}/{}_{b}\].