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 $\left( {0,0} \right)$ .