A) \[9\beta =2{{a}^{2}}\]
B) \[9\alpha =2{{\beta }^{2}}\]
C) \[2\alpha =9{{\beta }^{2}}\]
D) None of these
Correct Answer: B
Solution :
[b] Equation of tangent to the parabola \[{{y}^{2}}=4x\] Is\[y=mx+\frac{1}{m}\] ...(1) it passes through \[\left( \alpha ,\beta \right)\] \[\therefore \beta =m\alpha +\frac{1}{m}\] \[\Rightarrow {{m}^{2}}\alpha -m\beta +1=0\] let it roots are \[{{m}_{1}}\] and \[{{m}_{2}}\] according to condition if \[{{m}_{1}}=m\] them \[{{m}_{2}}=2m\] sum of roots \[{{m}_{1}}+{{m}_{2}}=m+2m=\frac{\beta }{\alpha }\] \[\Rightarrow 3m=\frac{\beta }{\alpha }\Rightarrow m=\frac{\beta }{3\alpha }\] ?....(2) Product of roots \[{{m}_{1}}.{{m}_{2}}=\left( m \right)\left( 2m \right)\] \[=\frac{1}{\alpha }\] \[\Rightarrow 2{{m}^{2}}=\frac{1}{\alpha }\] Form (1) \[\frac{2.{{\beta }^{2}}}{9{{\alpha }^{2}}}=\frac{1}{\alpha }\Rightarrow 2{{\beta }^{2}}=9\alpha \]
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