Question

If tangent drawn to a curve at a point is perpendicular to x-axis then at that point-
A. ${\displaystyle \frac{dy}{dx}}=0$
B. ${\displaystyle \frac{dx}{dy}}=0$
C. ${\displaystyle \frac{dy}{dx}}=1$
D. ${\displaystyle \frac{dy}{dx}}=-1$

Answer 1

Hint: A tangent line to a curve at a certain point is a line that touches the curve only at one point. Given that a tangent line drawn to a curve at a point is perpendicular to the x-axis. When a line is perpendicular to x-axis, it will be parallel to the y-axis. So, the x-coordinates of the line will not change only the y-coordinates will be changing. Use this info to solve the given question.

Complete step-by-step answer:
We are given that a tangent which is drawn to a curve at a point is perpendicular to x-axis.

We know that when a line is perpendicular to x-axis, it is parallel to y-axis and when a line is perpendicular to y-axis then it is parallel to x-axis.
We know that the slope of a line m is equal to $m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$ , where $x_1,x_2$ are the x-coordinates of the points of a line and $y_1,y_2$ are the y-coordinates of the points of the line.
Slope can also be written as $m={\displaystyle \frac{dx}{dy}}$ , where $dx$ is the change in x-coordinates and $dy$ is the change in y-coordinates.
But the tangent line is parallel to the y-axis, just its y-coordinates will be changing keeping the x-coordinates constant.
Therefore,
  $$\begin{gathered} dx=0 \\ \Rightarrow {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{dy}{0}} \\ \Rightarrow {\displaystyle \frac{dx}{dy}}={\displaystyle \frac{0}{dy}} \\ \therefore {\displaystyle \frac{dx}{dy}}=0 \end{gathered}$$
So, the correct option is Option B, ${\displaystyle \frac{dx}{dy}}=0$

So, the correct answer is “Option B”.

Note: Another approach
Straight line equation with slope m is $y=mx+c$
Differentiate the line equation with respect to x
 $$\begin{gathered} y=mx+c \\ {\displaystyle \frac{dy}{dx}}=m{\displaystyle \frac{dx}{dx}}+{\displaystyle \frac{d}{dx}}c \end{gathered}$$
c is a constant, so its differentiation will be zero.
 $$\begin{gathered} {\displaystyle \frac{dy}{dx}}=m\left(1\right)+0\left(\because {\displaystyle \frac{dx}{dx}}=1\right) \\ \Rightarrow {\displaystyle \frac{dy}{dx}}=m \end{gathered}$$
But for a line which is perpendicular to the x-axis the slope is infinity.
 $$\begin{gathered} \Rightarrow {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{1}{0}} \\ \Rightarrow {\displaystyle \frac{dx}{dy}}=0 \end{gathered}$$
Therefore, if tangent drawn to a curve at a point is perpendicular to x-axis then at that point ${\displaystyle \frac{dx}{dy}}=0$