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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, $C : y ^2= px ^3+ q$.....(1)
$\Rightarrow 2 y \frac{ dy }{ dx }=3 px ^2 \Rightarrow \frac{ dy }{ dx }=\frac{3 px ^2}{2 y }$.....(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, $(p-q)=2-(-7)=2+7=9$.