A) an ellipse of eccentricity \[\sqrt{3}\,/\,2\]
B) an ellipse of eccentricity \[1\,/\sqrt{3}\,\]
C) a hyperbola of eccentricity 2
D) an ellipse or a hyperbola depending on p
Correct Answer: A
Solution :
| \[2px+y\sqrt{1-{{p}^{2}}}=1\] | |
| i.e. \[y=-\frac{2p}{\sqrt{1-{{p}^{2}}}}x+\frac{1}{\sqrt{1-{{p}^{2}}}}\] | ?(i) |
| Let \[m=-\frac{2p}{\sqrt{1-{{p}^{2}}}}.\] Then \[1-{{p}^{2}}=\frac{4}{4+{{m}^{2}}}.\] | |
| \[\therefore \] equation of the line (i) becomes |
| \[y=mx+\frac{\sqrt{4+{{m}^{2}}}}{2}\] |
| i.e. \[y=mx+\sqrt{\frac{{{m}^{2}}}{4}+1}\] |
| \[\therefore \] the curve is an ellipse for which |
| \[{{b}^{2}}=\frac{1}{4}\] and \[{{a}^{2}}=1\] |
| \[\therefore \] \[{{e}^{2}}=\frac{3}{4}\] \[\therefore \] \[e=\frac{\sqrt{3}}{2}\] |
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