A) \[\left| m \right|\ge \frac{1}{2}\]
B) \[\left| m \right|\ge \frac{\sqrt{3}}{2}\]
C) \[\left| m \right|\ge 2\]
D) \[\left| m \right|\ge \frac{2}{3}\]
Correct Answer: B
Solution :
[b] If slope of tangent to hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is m then \[{{a}^{2}}{{m}^{2}}-{{b}^{2}}\ge 0\]. For given hyperbola \[{{\alpha }^{2}}{{m}^{2}}-{{({{\alpha }^{3}}+{{\alpha }^{2}}+\alpha )}^{2}}\ge 0\] \[\Rightarrow \,\,{{m}^{2}}\ge {{({{\alpha }^{2}}+\alpha +1)}^{2}}\Rightarrow {{m}^{2}}\ge {{\left[ {{\left( \alpha +\frac{1}{2} \right)}^{2}}+\frac{3}{4} \right]}^{2}}\Rightarrow {{m}^{2}}\ge \frac{3}{4}\]You need to login to perform this action.
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