Question

If $y=4x-5$ is tangent to the curve $y^{2}=px^{3}+q$ at $\left(\right.2, \, 3\left.\right)$ then $\left(\right.p,q\left.\right)$ is

• A. $\left(\right.2, \, 7\left.\right)$
• B. $\left(\right.-2, \, 7\left.\right)$
• C. $\left(\right.-2, \, -7\left.\right)$
• D. $\left(\right.2, \, -7\left.\right)$

Answer 1

Answer: (D)

Curve is $y^{2}=px^{3}+q$
$\therefore \, \, 2y\frac{d y}{d x}=3px^{2}$
$\Rightarrow \, \, \left(\frac{d y}{d x}\right)_{\left(\right. 2,3 \left.\right)}=\frac{3 p . 4}{2.3}$
$\Rightarrow \, \, \, 4=2p$
$\Rightarrow \, \, \, p=2$
Also, curve is passing through $\left(\right.2, \, 3\left.\right)$
$\therefore \, \, 9=8p+q$
$\Rightarrow \, \, \, q=-7$
$∴ \, \, \left(p , \, q\right)$ is $\left(2 , \, - 7\right)$