A) \[-2\le T\le 2\]
B) \[T\in (-\infty ,\,\,-8)\cup (8,\,\,\infty )\]
C) \[{{T}^{2}}<8\]
D) \[{{T}^{2}}\ge 8\]
Correct Answer: D
Solution :
Equation of the normal of the parabola\[{{y}^{2}}=4ax\]at the point\[(a{{t}^{2}},\,\,at)\]is \[y+tx=2at+a{{t}^{3}}\] ... (i) \[\because \]Eq. (i) cuts the parabola again at\[(a{{T}^{2}},\,\,2aT).\] Then, \[2aT+ta{{T}^{2}}=2at+a{{t}^{3}}\] \[\Rightarrow \] \[2a(T-t)=at({{T}^{2}}-{{t}^{2}})\] \[\Rightarrow \] \[2=-t(T+t)\] \[(\because \,\,t\ne T)\] \[\Rightarrow \] \[{{t}^{2}}+tT+2=0\] \[\therefore \,\,t\]is real,\[\therefore {{T}^{2}}-4\cdot 1\cdot 2\ge 0\] \[\Rightarrow \] \[{{T}^{2}}\ge 8\]
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