Question

What is the equation for a line that intersects the origin and is perpendicular to 2x – 4y = 13?
(a) y = -2
(b) y = 2x
(c) y = -2x
(d) \[y=\dfrac{1}{2}x\]
(e) \[y=\dfrac{1}{2}x-\dfrac{13}{4}\]

Answer 1

Hint: Rearrange the given equation similar to slope – intercept form of a line, y = mx + c. Find the value of m from it. The slope of line perpendicular is negative reciprocal of m. Substitute this m and point of origin (0, 0), get value c. Then find the required equation.

Complete step by step solution:
We have been given the equation of the line as,
\[\Rightarrow 2x-4y=13\]
Now let us rearrange the above expression as,
\[\Rightarrow 4y=2x-13\], divide the entire expression by 4.
\[\dfrac{4y}{4}=\dfrac{2x-13}{4}\Rightarrow y=\dfrac{x}{2}-\dfrac{13}{4}\] - (1)
We know that the slope – intercept form of the line is of the form,
y = mx + c – (2)
Now let us compare both equation (1) and (2).
Thus we can make out that slope = m = \[\dfrac{1}{2}\].
This line intersects the origin and it is said that it is perpendicular.
Thus the slope of the line perpendicular to this line will be the negative reciprocal i.e. m = \[\dfrac{-1}{m}\], put value of m in \[\dfrac{-1}{m}\].
Thus, \[\dfrac{-1}{m}=\dfrac{-1}{\dfrac{1}{2}}=-2\]
\[\therefore \] Slope of line perpendicular to this line will be -2.
Now let us put this value of slope in equation (2), we get,
y = -2x + c – (3)
It is said that the equation intersects or passes through the origin.
Hence let us substitute (x, y) = (0, 0) in equation (3).
Thus we get, \[0=-2\times 0+c\Rightarrow c=0\]
Let us put c = 0 in equation (3).
\[\begin{align}
  & y=-2x+c \\
 & y=-2x+0 \\
 & \Rightarrow y=-2x \\
\end{align}\]
Thus the equation of the required perpendicular line is,
y = - 2x
\[\therefore \] Option (c) is the correct answer.

Note: It is given in the question that the line is intersecting the origin, it means that the line passes through origin. So, we can directly write that the required form of the equation of line would be y = mx. We can also eliminate options on the same basis. Options (a) and (e) are not possible at all. Now, after obtaining the slope as m = -2, we can directly write the equation as y = -2x. Now, if we get the slope wrong by making any silly mistake like change in sign or not taking reciprocal, we will get a different option as the right answer.