First slide

Ellipse

Question

Which of the following equations ( t being the parameter ) can't represent a hyperbola

Difficult

A

$\frac{t x}{a} - \frac{y}{b} + t = 0, \frac{x}{a} + \frac{t y}{b} - 1 = 0$
B

$x = \frac{a}{2} (t + \frac{1}{t}), y = \frac{b}{2} (t - \frac{1}{t})$
C

$x = e^{t} + e^{- t}, y = e^{t} - e^{- t}$
D

$x^{2} = 2 (\cos t + 3), y^{2} = 2 (\cos^{2} \frac{t}{2} - 1)$

Solution

Eliminating t.
By Solving given equations, in 1^st point we have
$x = a (\frac{1 - t^{2}}{1 + t^{2}}), y = b (\frac{2 t}{1 + t^{2}})$
If $t = \tan θ$ then $\frac{x}{a} = \cos 2 θ, \frac{y}{5} = \sin 2 θ$
$\Rightarrow \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is an ellipse, not the hyperbola

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