Question

The equation of the tangents at $(2,3)$ on the curve $y^2=a{x}^3+b$ is $y=4x-5$ find the value of $a$ & $b$.

• A. $2$ and $-2$
• B. $4$ and $-7$
• C. $2$ and $-7$
• D. None of these

Answer 1

The correct option is C $2$ and $-7$

equation of curve :- $y^2=a{x}^3+b$

difference with respect to x

$\therefore 2y\frac{dy}{dx}=3a{x}^2$

$\therefore \frac{dy}{dx}=\frac{3a{x}^2}{2y}$$\left(\frac{dy}{dx}\right)(x=2,y=3)=\frac{3a(2{)}^2}{2\times 3}$

$\frac{dy}{dx}=2a$

$\therefore$equation of tangent:-

$\therefore y-3=2a(x-2)$

$\therefore y=2ax-4a+3$

$\therefore 2a=4$

$\therefore a=2$

Now,

$3^2=a(2{)}^3+b$

$\therefore 9=16+b$

$\therefore b=5-16=-7$

$\therefore b=-7$.