Conic-Sections Question 595
Question: The curve described parametrically by $ x=2-3\sec t,y=1+4\tan t $ represents:
Options:
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} $
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Answer:
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 9e^{2}=25\Rightarrow e^{2}=\frac{25}{9}\Rightarrow e=\frac{5}{3} $
BETA
