Question

How do you write an equation in slope intercept form perpendicular to \[y = \dfrac{1}{{2x - 4}}\], contains $\left( {4,1} \right)$?

Answer 1

Hint: Given that the two lines are perpendicular to each other, therefore we can find the slope of the unknown line using the slope of the line given in the formula.
Once we’ve found the slope $m$, we can substitute the known values in the equation $y = mx + c$ to find the value of $c$ and thus solve the question.

Formula used: $y = mx + c$,
If two lines are perpendicular to each other than the products of their slopes is $ - 1$ i.e. ${m_1} \times {m_2} = - 1$ , where ${m_1}$ and ${m_2}$ are slopes of the respective lines.

Complete step-by-step solution:
Given the equation is: $y = \dfrac{1}{{2x}} - 4$, this is in the form of the slope intercept formula.
Thus, we can compare it with the generic equation $y = mx + c$ to get our first slope ‘m’.
Here $m = \dfrac{1}{2}$
Now, we need to find the slope of the unknown line which is perpendicular to the given line.
Let us consider the slope of the unknown perpendicular line to be ${m_1}$
As we know that the product of slopes of two lines perpendicular to each other is $ - 1$
Therefore, $m \times {m_1} = - 1$ , where $m = \dfrac{1}{2}$ and ${m_1}$ is the unknown quantity
Putting in the values of the known variable in the above formula, we get:
$\dfrac{1}{2} \times {m_1} = - 1$
On cross multiplying, we get:
$ \Rightarrow {m_1} = - 2$
Therefore the slope of the unknown line = $ - 2$
Now, the unknown line passes through the points ($4,1$) and the line is to be given in the slope-intercept form
Thus, the slope intercept form is given as: $y = mx + c$, where m= slope, y and x are the coordinates and c is the intercept.
Now, in order to find the value of ‘c’, we substitute the values of slope with ${m_1}$ and the values of coordinates x and y with $\left( {4,1} \right)$
Thus, we get: $y = {m_1}x + c$
On substituting the values in the above equation, we get:
$ \Rightarrow 1 = - 2 \times 4 + c$
On simplifying, we get:
$1 = - 8 + c$
Now, we bring the variable to the left side and keep the numbers on the right hand side. As we change the sides of the numbers and the variables, therefore their respective signs also change:
\[ \Rightarrow - c = - 8 - 1\]
On further simplifying, we get:
$ \Rightarrow - c = - 9$
On multiplying both sides of the equation with a ‘—‘, we get:
$ + c = + 9$ , since if we multiply two negatives, then it gives us a positive $\left( - \right) \times \left( - \right) = \left( + \right)$
Thus, $c = 9$
The equation for the unknown line is given as: $y = {m_1}x + c$
We substitute the values of $c$ and ${m_1}$ in the above equation to get the required answer:
$y = - 2x + 9$

The other equation of line is y=-2x+9.

Note: If two slopes ${m_1}$ and ${m_2}$ are perpendicular to each other, then the product of their slopes is $ - 1$ but if two slopes are parallel to each other then ${m_1} = {m_2}$.