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Q.
If $y=4 x-5$ is a tangent to the curve $C: y^{2}=p x^{3}+q$ at $M$ $(2,3)$ then the value of $(p-q)$ is
Application of Derivatives
Solution:
Given, $C: y^{2}=p x^{3}+q$ ...(1)
$\Rightarrow 2 y \frac{d y}{d x}=3 p x^{2}$
$\Rightarrow \frac{d y}{d x}=\frac{3 p x^{2}}{2 y}$ ...(2)
Put $x=2$ and $y=3$ in (1), we get $9=8 p+ q$ ...(3)
Also, from (2), we get $\left.\frac{d y}{d x}\right]_{M(2,3)}=\frac{3 p(4)}{6}=2 p=4$ ....(4)
$\therefore $ From (2) and (3), we get $p=2, q=-7$.
Hence, $(p-q)=2-(-7)=2+7=9$.