A) An ellipse centred at (2, 1) and of eccentricity \[\frac{3}{5}\]
B) A circle centred at (2, 1) and of radius 5 units
C) A hyperbola centred at (2, 1) & of eccentricity \[\frac{8}{5}\]
D) A hyperbola centred at \[(2,1)\]& of eccentricity \[\frac{5}{3}\]
Correct Answer: D
Solution :
[d] Given, \[x=2-3\sec t,y=1+4\tan t\] \[\Rightarrow \sec t=\frac{x-2}{-3},\tan t=\frac{y-1}{4}\] Since, \[{{\sec }^{2}}t-{{\tan }^{2}}t=1\] \[\therefore \frac{{{(x-2)}^{2}}}{9}-\frac{{{(y-1)}^{2}}=1}{16},\] Which is a hyperbola with centre at (2, 1) and eccentricity e, given by \[16=9({{e}^{2}}-1)\] \[\Rightarrow 9{{e}^{2}}=25\Rightarrow {{e}^{2}}=\frac{25}{9}\Rightarrow e=\frac{5}{3}\]
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