Question

x = 6 cosec theta ,y = 8 cot theta​. ( eliminate theta from the following)

Answer 1

Answer:

$\bf{\frac{x^2}{36}-\frac{y^2}{64}=1}$

Step-by-step explanation:

$\text{Formula used:}$

$cosec^2A-cot^2A=1$

$\text{Given:}$

$x=6\:cosec\theta\:\text{and}\:y=8\:cot\theta$

$\implies\:\frac{x}{6}=cosec\theta\:\text{and}\:\frac{y}{8}=cot\theta$

$\text{we know that}\:cosec^2\theta-cot^2\theta=1$

$\implies\:(\frac{x}{6})^2-(\frac{y}{8})^2=1$

$\implies\:\frac{x^2}{36}-\frac{y^2}{64}=1$

Answer 2

Answer:  The required equation after eliminating $\theta$ is $\dfrac{x^2}{36}-\dfrac{y^2}{64}=1.$

Step-by-step explanation:  We are given to eliminate $\theta$ from the following equations :

$x=6\csc\theta~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\y=8\cos\theta~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

We will be using the following trigonometric identity :

$\csc^2x-\cot^2x=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

From equation (i), we have

$x=6\csc\theta\\\\\Rightarrow \dfrac{x}{6}=\csc\theta\\\\\\\Rightarrow \left(\dfrac{x}{6}\right)^2=\csc^2\theta~~~~~~~~~~~~~[\textup{Squaring both sides}]\\\\\\\Rightarrow \dfrac{x^2}{36}=\csc^2\theta~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iv)$

From equation (ii), we get

$y=8\cot\theta\\\\\Rightarrow \dfrac{y}{8}=\cot\theta\\\\\\\Rightarrow \left(\dfrac{y}{8}\right)^2=\cot^2\theta~~~~~~~~~~~~~[\textup{Squaring both sides}]\\\\\\\Rightarrow \dfrac{y^2}{64}=\cot^2\theta~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(v)$

Subtracting equation (v) from equation (iv), we get

$\dfrac{x^2}{36}-\dfrac{y^2}{64}=\csc^2\theta-\cot^2\theta\\\\\\\Rightarrow \dfrac{x^2}{36}-\dfrac{y^2}{64}=1~~~~~~~~~~~[\textup{Using identity (iii)}]$

Thus, the required equation after eliminating $\theta$ is $\dfrac{x^2}{36}-\dfrac{y^2}{64}=1.$