Application Of Derivatives Ques 7
7. If the normal to the curve $y=f(x)$ at the point $(3,4)$ makes an angle $\frac{3 \pi}{4}$ with the positive $X$-axis, then $f^{\prime}(3)$ is equal to
(2000, 1M)
(a) -1
(b) $-3 / 4$
(c) $4 / 3$
(d) 1
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Answer:
Correct Answer: 7.(d)
Solution:
Formula:
Equation of tangent and normal :
- Slope of tangent $y=f(x)$ is $\frac{d y}{d x}=f^{\prime}(x)_{(3,4)}$
Therefore, slope of normal
$=-\frac{1}{f^{\prime}(x)_{(3,4)}} =-\frac{1}{f^{\prime}(3)} $
$\text { But } -\frac{1}{f^{\prime}(3)} =\tan (\frac{3 \pi}{4}) [given] $
$\Rightarrow \frac{-1}{f^{\prime}(3)} =\tan (\frac{\pi}{2}+\frac{\pi}{4})=-1 $
$\therefore f^{\prime}(3) =1 $
BETA
