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  • Need solution for RD Sharma Maths Class 12 Chapter 15 Tangents and Normals Excercise Fill in the blanks Question 13

Need solution for RD Sharma Maths Class 12 Chapter 15 Tangents and Normals Excercise Fill in the blanks Question 13

Answers (1)

Answer:

                f^{\prime}(3)=1

Hint:

First find the slope of given curve at point \left ( 3,4 \right ) , then compare with\frac{3\pi}{4}

Given:

Given curve,

                y=f(x)

To find:

We have to find {f}'\left ( 3 \right ) , if the normal to the given curve at \left ( 3,4 \right ) makes an angle \frac{3\pi}{4} with positive x-axis.

Solution:

Here we have,

                y=f(x)

On differentiating with respect to x , we get

\Rightarrow \quad \frac{d y}{d x}=f^{\prime}(x)

 Slope of tangent at  (3,4)=\frac{d y}{d x}=f^{\prime}(x)_{(3,4)}

Therefore, slope of normal

\begin{aligned} &=\frac{-1}{f^{\prime}(x)_{(3.4)}} \\ &=\frac{-1}{f^{\prime}(3)} \end{aligned}

But  \frac{-1}{f^{\prime}(3)}=\tan \left(\frac{3 \pi}{4}\right)      [Given]

\begin{aligned} &\Rightarrow \quad \frac{-1}{f^{\prime}(3)}=\tan \left(\frac{\pi}{2}+\frac{\pi}{4}\right) \\\\ &\Rightarrow \quad \frac{-1}{f^{\prime}(3)}=-1 \end{aligned}

 

Hence   f^{\prime}(3)=1

 

Posted by

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