| DIRECTION: Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Choose the correct answer (Only one option is correct) from the following - |
| Statement-1: Through \[(1,\lambda +1),\]there cannot be more than one-normal to the parabola \[{{y}^{2}}=4x\]if \[\lambda <2.\] |
| Statement-2: The point \[(1,\lambda +1),\]lies out side the parabola for all \[\lambda \ne 1.\] |
A) Statement-1 is false, Statement-2 is true.
B) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D) Statement-1 is true, Statement-2 is false.
Correct Answer: C
Solution :
Any normal to the parabola \[{{y}^{2}}=4x\]is \[y+tx=2t+{{t}^{3}}\]If this passes through \[(\lambda ,\lambda +1),\], then \[{{t}^{3}}+t(2-\lambda )-\lambda -1=0=f(t)\] (say) \[\lambda <2\]then \[f'(t)=3{{t}^{2}}+(2-\lambda )>0\]\[\Rightarrow f(t)=0\] will have only one real root \[\Rightarrow \]is true The Statement-2 is also true since \[{{(\lambda -1)}^{2}}>4\lambda \] is true for all \[\lambda \ne 1\]. The Statement-2 is true but does not follow true Statement-2.
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