A) \[u={{e}^{x}}\sin x\]
B) \[v={{e}^{x}}\cos x\]
C) \[v\frac{du}{dx}-u\frac{dv}{dx}={{u}^{2}}+{{v}^{2}}\]
D) \[\frac{{{d}^{2}}u}{d{{x}^{2}}}=2v\]
Correct Answer: B
Solution :
The equation of any tangent to \[{{y}^{2}}=8ax\] is \[y=mx+\frac{2a}{m}\] ?(1) If it touches \[{{x}^{2}}+{{y}^{2}}=2{{a}^{2}},\]then \[{{\left( \frac{2a}{m} \right)}^{2}}=2{{a}^{2}}(1+{{m}^{2}})\] \[[\because {{c}^{2}}={{a}^{2}}(1+{{m}^{2}})]\] \[\Rightarrow \]\[2={{m}^{2}}({{m}^{2}}+1)\] \[\Rightarrow \]\[{{m}^{4}}+{{m}^{2}}-2=0\] \[\Rightarrow \]\[({{m}^{2}}+2)({{m}^{2}}-1)=0\]\[\Rightarrow \]\[{{m}^{2}}-1=0\] \[\Rightarrow \]\[m=\pm 1\]\[\Rightarrow \]\[\Rightarrow \]\[\Rightarrow \] Putting the values of m in Eq. (i), we get \[y=\pm \](x + 2a) as the equations of common tangents.
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