question_answer

The point of contact of the line \[y=x-1\] with\[3x-4{{y}^{2}}=12\]is

A) \[(4,\,\,3)\]                        

B) \[(3,\,\,4)\]

C) \[(4,\,\,-3)\]                      

D)  None of these

Correct Answer: A

Solution :

The equations of the line and hyperbola are                 \[y=x-1\]                                             ... (i)                 \[3{{x}^{2}}-4{{y}^{2}}=12\]                                        ... (ii) From Eqs. (i) and (ii), we get                 \[3{{x}^{2}}-4{{(x-1)}^{2}}=12\] \[\Rightarrow \]               \[3{{x}^{2}}-4({{x}^{2}}-2x+1)=12\] or            \[{{x}^{2}}-8x+16=0\Rightarrow x=4\] From Eq. (i),       \[y=3\] Hence, required point of contact is\[(4,\,\,3)\].