A) \[x-2y+4=0;\text{ }x+2y+4=0\]
B) \[2x-y+4=0;2x+y+4=0\]
C) \[2x-y+2=0;2x+y+2=0\]
D) None of these
Correct Answer: A
Solution :
[a] Ellipse \[\frac{{{x}^{2}}}{8}+\frac{{{y}^{2}}}{2}=1\] Equation of tangent \[y=mx\pm \sqrt{8{{m}^{2}}+2}\] ... (1) and parabola \[{{y}^{2}}=4x\] Equation of tangent \[\Rightarrow y=mx+\frac{1}{m}\] ... (2) for common tangent \[\pm \sqrt{8{{m}^{2}}+2}=\frac{1}{m}\] \[\Rightarrow 8{{m}^{2}}+2=\frac{1}{{{m}^{2}}}\] \[\Rightarrow 8{{m}^{4}}+2{{m}^{2}}-1=0\] \[\Rightarrow \left( 4{{m}^{2}}-1 \right)\left( 2{{m}^{2}}+1 \right)=0\Rightarrow m=\pm \frac{1}{2}\] \[\therefore E{{q}^{n}}\] of common tangent \[y=\pm \frac{1}{2}x\pm 2\] \[\Rightarrow x-2y+4=0\text{ }and\text{ }x+2y+4=0\]
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