Question

Find all points on the curve that have the given slope.x = 2cost, y= 8sint, slope =-1

Answer 1

Answer:

$t_{1} = 0.422\pi \pm \pi\cdot i$, $\forall \,i \in \mathbb{N}_{O}$

$t_{2} = 1.422\pi \pm \pi\cdot i$, $\forall \,i \in \mathbb{N}_{O}$

Step-by-step explanation:

In the case of parametric equations, the slope of the curve is equal to:

$\frac{dy}{dx} = \frac{\frac{dy}{dt} }{\frac{dx}{dt} }$

Where $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are the first derivatives of $x$ and $y$ regarding $t$. Let be $x(t) =2\cdot \cos t$ and $y(t) = 8\cdot \sin t$, their first derivatives are found:

$\frac{dx}{dt} = -2\cdot \sin t$ and $\frac{dy}{dt} = 8\cdot \cos t$

Thus, equation for the slope is:

$\frac{dy}{dx} = -\frac{8\cdot \cos t}{2\cdot \sin t}$

$\frac{dy}{dx} = -\frac{4}{\tan t}$

If $\frac{dy}{dx} = -1$, then:

$-1 = -\frac{4}{\tan t}$

$\tan t = 4$

Tangent is positive at 1st quadrant and is a function with a periodicity of $\pi$, the set of solutions are:

$t_{1} = 0.422\pi \pm \pi\cdot i$, $\forall \,i \in \mathbb{N}_{O}$

$t_{2} = 1.422\pi \pm \pi\cdot i$, $\forall \,i \in \mathbb{N}_{O}$