Question

Find the equation of a line which slope 2 and the length of the perpendicular form the origin is \[\sqrt 5 \] .

Answer 1

Hint: The general equation of a line is given as \[y = mx + c\] , where \[m\] is the slope of the line and \[c\] is the y-intercept.
In this question we are given with a slope of a line so by using the general equation of a straight line we will find the equation of a straight line and also we are given with perpendicular distance of the origin from the straight line so using the relation we will find the value of y-intercept \[c\] and by substituting the y-intercept we will find the equation of the line.

Complete step-by-step answer:
Given the slope of the line is \[m = 2\]
Now as we know the general equation of a line is given as \[y = mx + c\] , hence by substituting the values of slope we can write the equation as
 \[y = 2x + c\]
This equation can also be written as
 \[2x - y + c = 0\]
Now we are given with the length of the perpendicular form the origin as \[\sqrt 5 \] , the coordinate of the origin is \[\left( {0,0} \right)\]
We know the perpendicular distance of a line from a point is given by the formula \[d = \pm \left( {\dfrac{{a{x_1} + b{y_1} + c}}{{\sqrt {{a^2} + {b^2}} }}} \right)\]
Hence by substituting the values of the equation of the line and the coordinate of the origin we can write,
 \[
  \sqrt 5 = \pm \left( {\dfrac{{a\left( 0 \right) + b\left( 0 \right) + c}}{{\sqrt {{{\left( 2 \right)}^2} + {{\left( 1 \right)}^2}} }}} \right) \\
   \Rightarrow \sqrt 5 = \left( {\dfrac{{0 + 0 + c}}{{\sqrt {4 + 1} }}} \right) \\
   \Rightarrow c = \sqrt 5 \times \sqrt 5 \\
   \Rightarrow c = 5 \;
 \]
Hence by substituting the value of \[c\] in equation (i) we can write
 \[2x - y + 5 = 0\]
Therefore the equation of a line is \[2x - y + 5 = 0\]
So, the correct answer is “ \[2x - y + 5 = 0\] ”.

Note: The perpendicular distance of a line from a point is given by the formula \[d = \pm \left( {\dfrac{{a{x_1} + b{y_1} + c}}{{\sqrt {{a^2} + {b^2}} }}} \right)\] , where equation of the line is \[ax + by + c = 0\] and the coordinate of the point is \[\left( {{x_1},{y_1}} \right)\] . In this question since the perpendicular line was from the origin so we took the coordinate of the point as \[\left( {0,0} \right)\] .