Question

Please anyone assist me solve those two mathematics questions.

Answer 1

Explanation:

(i) We know that
`x = rcos(\theta)\:\text{and}\:y = rsin(\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 = (5)/((\cos^2\theta - \sin^2\theta))`

`\:\:\:\:\:=(5)/(cos(2\theta)) = 5\sec{2\theta}`

(ii) Recall that

`\cot{\theta} = (cos(\theta))/(sin(\theta))`

`\csc{\theta} = (1)/(sin(\theta))`

so we can write r as

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

`\:\:\:\:= 8\left((cos(\theta))/(sin(\theta))\right)\left((1)/(sin(\theta))\right)`

`\:\:\:\:= 8\left((x)/(y)\right)\!\left((r)/(y)\right) = (8xr)/(y^2)`

Upon simplification, we get

`y^2 = 8x`

Answer 2

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

2) i am stuck on this one (sorry)