A) \[p=2,q=-7\]
B) \[p=-2,q=7\]
C) \[p=-2,q=-7\]
D) \[p=2,q=7\]
E) \[p=0,q=7\]
Correct Answer: A
Solution :
The equation of curve is \[{{y}^{2}}=p{{x}^{3}}+q\] \[\therefore \] \[2y\frac{dy}{dx}=\frac{3p{{x}^{2}}}{2y}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{3p{{x}^{2}}}{2y}\] \[\therefore \] \[{{\left( \frac{dy}{dx} \right)}_{(2,3)}}=\frac{3p{{(2)}^{2}}}{2.3}=2p\] The equation of tangent at (2, 3) is \[(y-3)=2p(x-2)\] \[\Rightarrow \] \[2px-y=4p-3\] ...(i) This is similar to\[y=4x-5\] \[\therefore \] \[2p=4\]and\[4p-3=5\] \[\Rightarrow \] \[p=2\]and\[p=2\] The point (2, 3) lies on the curve. \[\therefore \] \[9=8p+q\] \[\Rightarrow \] \[9=16+q\] (\[\because \]\[p=2\]) \[\Rightarrow \] \[q=-7\]
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