A) \[(\pm 4,\,8)\]
B) \[(\pm 1,1/2)\]
C) \[(\pm 2,2)\]
D) None of these
Correct Answer: C
Solution :
Let \[(x,y)\] be any point on the parabola \[2y={{x}^{2}}.\] Let \[f(x)={{(x-0)}^{2}}+{{(y-3)}^{2}}\] \[={{x}^{2}}+\left( \frac{{{x}^{2}}}{2}-3 \right)\] \[\Rightarrow \] \[f'(x)=2x+2{{\left( \frac{{{x}^{2}}}{2}-3 \right)}^{2}}\] Put \[f'(x)=0\] \[\Rightarrow \] \[2x\left( 1+\frac{{{x}^{2}}}{2}-3 \right)=0\] Now, \[f''(x)=2+2\left( \frac{{{x}^{2}}}{2}-3 \right)+2x(x)\] At \[x=0,\,f''(x)=2-6=-4<0,\] maximum At \[x=+2,\,f''(x)>0,\] minimum At \[x=-2,\,\,f''(x)>0,\] minimum \[\therefore \] At \[x=\pm 2\] \[\Rightarrow \] \[y=2\] \[\therefore \] Required point is \[(\pm 2,2)\].
You need to login to perform this action.
You will be redirected in
3 sec
