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  4. If y =4 x -5 is tangent to the curve y 2= px 3+ q at (2,3) then

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

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Solution:

$2 y \frac{d y}{d x}=3 px ^{2} ; \frac{d y}{d x} \quad(2,3)=2 p ;$
equating slopes $2 p =4 $
$\Rightarrow p =2 ;$
since $(2,3)$ lies on the curve
$y ^{2}= px ^{3}+ q ;(3)^{2}$
$= p (2)^{3}+ q $
$\Rightarrow q =8 p + q $
$\Rightarrow 9=16+ q$
$ \Rightarrow q =-7$
$= p (2)^{3}+ q$
$\Rightarrow q =8 p + q $
$\Rightarrow 9=16+ q$
$ \Rightarrow q =-7$