Question

The normal to a given curve is
parallel to x-axis If:
O dx/dy=1
O dy/dx=0
O dx/dy=0
O dy/dx=1​

Answer 1

To find:

Condition when the normal to a given curve is zero.

Calculation:

Let us consider a function f(x) , such that slope of tangent be $m_{t}$ and slope of normal be $m_{n}$.

Now , we know that , normal is perpendicular to tangent, such that:

$m_{n}\times m_{t}= -1$

$=> m_{n} \times \dfrac{dy}{dx} = -1$

$=> m_{n} = - \dfrac{dx}{dy}$

Now , as per the question , the normal is parallel to x axis , so slope of normal is equal to slope of x axis (i.e. zero)

$\therefore m_{n}= 0$

$=>- \dfrac{dx}{dy}=0$

$=> \dfrac{dx}{dy} = 0$

So, final answer is :

$\boxed{\bf{\large{\dfrac{dx}{dy} = 0}}}$

Answer 2

Given :  normal to a given curve is  parallel to x-axis

To Find : Choose correct option :

O dx/dy=1

O dy/dx=0

O dx/dy=0

O dy/dx=1​

Solution:

Normal to a  curve is perpendicular to the tangent to the curve

Tangent to curve is defined by  dy/dx  

normal to a given curve is  parallel to x-axis  

Hence Tangent will be parallel to y axis

=> slope of line parallel to y axis  =  1/0

=>  dy/dx   = 1/0

=>  dx/dy  = 0

dx/dy=0 is the correct option

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