Applications Of Derivatives Question 13
Question: If $ x+4y=14 $ is a normal to the curve $ y^{2}=ax^{3}-\beta $ at (2, 3), then the value of $ \alpha +\beta $ is
Options:
A) 9
B) $ -,5 $
C) 7
D) $ -,7 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ y^{2}=\alpha x^{3}-\beta $ or $ \frac{dy}{dx}=\frac{3\alpha x^{2}}{2y} $ Therefore, slope of the normal at (2, 3) is $ {{( -\frac{dx}{dy} )}_{(2,3)}}=-\frac{2\times 3}{3\alpha {{(2)}^{2}}}=-\frac{1}{2\alpha }=-\frac{1}{4} $ Or $ \alpha =2 $ Also, (2, 3) lies on the curve. Therefore, $ 9=8\alpha -\beta $ or $ \beta =16-9=7 $ or $ \alpha +\beta =9. $
BETA
