Question

What is the formula of a line that is perpendicular to $y = \dfrac{1}{3}x + 9$ and includes the point $\left( {3,4} \right)$ ?
(A) $y = \dfrac{1}{3}x + 5$
(B) $y = - \dfrac{1}{3}x + 13$
(C) $y = 3x + 5$
(D) $y = - 3x + 5$
(E) $y = - 3x + 13$

Answer 1

Hint: In the given problem, we are required to find the equation of a line that is perpendicular to a given line and contains the point provided to us. We first find the slope of the line whose equation is given to us. As we know, the product of slopes of perpendicular lines is -1. Using this relation slope of the required line can be obtained. Then, finally we substitute the coordinates of the point given to us in the slope point form of a straight line.
Formulae used.
$\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where $\left( {{x_1},{y_1}} \right)$
This is slope point form of the line where ,
$x_1$ and $y_1$ are X and Y coordinates of points lying on a given line.

Complete step-by-step answer:
So, we are given the straight line $y = \dfrac{1}{3}x + 9$.
We know that the slope-intercept form of a straight line is $y = mx + c$ where m is the slope of the line and c is the y-intercept of the line.
So, comparing the equation with the slope point form of the line, we get the slope as
$m = \dfrac{1}{3}$.
Now, we know that the product of slopes of two perpendicular lines is always equal to $ - 1$. Hence, let us assume the slope of the required line as t.
Then, we have,
$t\left( {\dfrac{1}{3}} \right) = - 1$
Multiplying both sides of the equation by $3$, we get,
$ \Rightarrow t = - 3$
So, the slope of the required line is $ - 3$.
Now, we are also given the coordinates of a point lying on the line as $\left( {3,4} \right)$.
We know the slope point form of the line, where we can find the equation of a straight line given the slope of the line and the point lying on it. The slope point form of the line can be represented as: $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where $\left( {{x_1},{y_1}} \right)$ is the point lying on the line given to us and m is the slope of the required straight line.
Considering ${x_1} = 3$ and ${y_1} = 4$, we get,
Therefore, the required equation of line is as follows:
$ \Rightarrow \left( {y - 4} \right) = - 3\left( {x - 3} \right)$
On opening the brackets and simplifying further, we get,
$ \Rightarrow y - 4 = - 3x + 9$
Taking all the variables to the left side of the equation and all the constant on the right side of equation, we get,
$ \Rightarrow y + 3x = 4 + 9$
Simplifying the equation, we get,
$ \Rightarrow y = - 3x + 13$
Hence, the equation of the required straight line is: $y = - 3x + 13$. So, option (E) is the correct answer.
So, the correct answer is “Option E”.

Note: The slope of a line is a measure of the incline of the line. The slope is a number which indicates both the direction of the line and the steepness of the line. It can be calculated as the tangent of inclination of the line made with the positive direction of the X axis. Slope of the X axis is 0 and that of the Y axis is infinite.