Question

Please anyone assist me solve those two mathematics questions.

Answer 1

Answer:

1) The polar form of that equation will be r= √5cos(2θ)

2) i am stuck on this one (sorry)

Answer 2

Step-by-step explanation:

(i) We know that $x = r\cos{\theta}\:\text{and}\:y = r\sin{\theta}.$ So we can write the given equation as

$r^2\cos^2\theta - r^2\sin^2\theta = 5$

$\Rightarrow r^2(\cos^2\theta - \sin^2\theta) = 5$

$r^2 = \dfrac{5}{(\cos^2\theta - \sin^2\theta)}$

$\:\:\:\:\:=\dfrac{5}{\cos{2\theta}} = 5\sec{2\theta}$

(ii) Recall that

$\cot{\theta} = \dfrac{\cos{\theta}}{\sin{\theta}}$

$\csc{\theta} = \dfrac{1}{\sin{\theta}}$

so we can write r as

$r = 8\cot{\theta}\csc{\theta}$

$\:\:\:\:= 8\left(\dfrac{\cos{\theta}}{\sin{\theta}}\right)\left(\dfrac{1}{\sin{\theta}}\right)$

$\:\:\:\:= 8\left(\dfrac{x}{y}\right)\!\left(\dfrac{r}{y}\right) = \dfrac{8xr}{y^2}$

Upon simplification, we get

$y^2 = 8x$