If the equetion of tangent to the curve `y^(3)=ax^(3)+` b at point (2,3) is y = 4x = 5, then find the values of a and b.

Options

A. `a=2, b=7`
B. `a=7, b=2`
C. `a=2, b= -7`
D. `a= -2,b=7`

Answer

C

Explanation

The equation of the curve is `y^(2)=ax^(3)+b.`
`therefore 2y (dy)/(dx)=3ax^(2) rArr (dy)/(dx) = (3ax^(2))/(2y) rArr ((dy)/(dx))_((2","3))=2a`
The equation of the tangent at (2, 3) is
`y-3=2a(x-3) or, 2ax-y-6a+3=0`
But, the equation of the tangent at (2, 3) is `y=4x-5.`
Comparing these two equations, we get
`2a=4 and -6a+3= -5 rArr a=2`
As (2, 3) lies on the curve `y^(2)=ax^(3)+b.`
`therefore 9=8a+b rArr 9=16+b rArr b = -7`