Question

If the line $y=4x-5$ touches to the curve $y^2=a{x}^3+b$ at the point $(2,3)$ then $7a+2b=$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

Answer: (A)

$0$

The slope of given line $y=4x-5$ is $m=4$.

Now, since the given line touches the curve at $(2,3)$, the slope of curve $\frac{dy}{dx}$ at the given point is equal to the slope $m$ of the line.

Hence, for slope of curve, differentiating both sides w.r.t x:-

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

Putting the value $x=2$ and $y=3$

$6\frac{dy}{dx}=3a\times 4⟹6\times 4=12a$

$⟹a=2$

Now, the given point lies on the curve, so it must satisfy the equation of the curve:-

$⟹(3{)}^2=2\times (2{)}^3+b$

$⟹b=-7$

$7a+2b=(7\times 2-2\times 7)=0$

Hence, answer is option-(A).