Question

If $y=4x-5$ is a tangent to the curve $C:y^2=p{x}^3+q$ at $M$ $(2,3)$ then the value of $(p-q)$ is

Answer 1

Answer: 9

Given, $C:y^2=p{x}^3+q$ ...(1)
$\Rightarrow 2y\frac{dy}{dx}=3p{x}^2$
$\Rightarrow \frac{dy}{dx}=\frac{3p{x}^2}{2y}$ ...(2)
Put $x=2$ and $y=3$ in (1), we get $9=8p+q$ ...(3)
Also, from (2), we get ${\frac{dy}{dx}]}_{M(2,3)}=\frac{3p(4)}{6}=2p=4$ ....(4)
$\therefore$ From (2) and (3), we get $p=2,q=-7$.
Hence, $(p-q)=2-(-7)=2+7=9$.