Question

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

• A. $p=2,q=-7$
• B. $p=-2,q=7$
• C. $p=-2,q=-7$
• D. $p=2,q=7$

Answer 1

Answer: (A)

$p=2,q=-7$

Given equation of curve
$y^2=p{x}^3+q$
$\frac{dy}{dx}=\frac{3p{x}^2}{y}$
Slope of tangent at (2,3) $=(\frac{dy}{dx}{)}_{(2,3)}=2p$
Given equation of tangent is
$y=4x-5$
Slope of tangent =4
$\Rightarrow 2p=4$
$\Rightarrow p=2$

Since, (2,3) lies on

$y^2=p{x}^3+q$

$9=16+q$
$\Rightarrow q=-7$