Question

Eliminate z, if x=a cosec z
Y = b cot z​
and
ans

Answer 1

$x = a \csc(z) \\ \csc(z) = \frac{x}{a}$
Squaring both sides,
${ \csc \: }^{2}z = \frac{ {x}^{2} }{ {a}^{2} }$
-----(i)

We have,
$y = b \cot(z) \\ \cot(z) = \frac{y}{b}$
Squaring both sides,

${ \cot }^{2} z = \frac{ {y}^{2} }{ {b}^{2} }$
------(ii)



Since we have the following relation,
$1 + { \cot}^{2}z = { \csc}^{2} z$
Therefore,

subsituting (i) and (ii) here,

$1 + \frac{ {y}^{2} }{ {b}^{2} } = \frac{ {x}^{2} }{ {a}^{2} }$
Solving it furthur,
$1 = \frac{ {x}^{2} }{ {a}^{2} } - \frac{ {y}^{2} }{ {b}^{2} } \\ \frac{ {b}^{2} {x}^{2} - {a}^{2} {y}^{2} }{ {a}^{2} {b}^{2} } = 1 \\ \\ {b}^{2} {x}^{2} - {a}^{2} {y}^{2} = {a}^{2} {b}^{2}$



Hence, Eliminated successfully.