Question

1. give parametrizations for the following:

(a) a line passing through the points (0,2,4) and (3, - 1, - 6) in that order.
(b) a circle with center (2,4, - 3) and radius 5 which is parallel to the xy - plane. give this circle a counter - clockwise orientation when viewed from \above.\ (i.e. a counter - clockwise orientation when viewed from a point with a large z - value.)
(c) the ellipse (either orientation): $\frac{x^{2}}{36}+\frac{y^{2}}{25}=1$.
(d) the ellipse (either orientation): $\frac{(x - 1)^{2}}{36}+\frac{(y + 2)^{2}}{25}=1$

Explanation:

Step1: Find the parametric - form of a line

The parametric equations of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ are given by $x = x_1+(x_2 - x_1)t$, $y = y_1+(y_2 - y_1)t$, where $t\in R$. Here, $(x_1,y_1)=(0,2,4)$ and $(x_2,y_2)=(3, - 1,-6)$. So, $x = 0+(3 - 0)t=3t$, $y = 2+( - 1 - 2)t=2 - 3t$, $z = 4+( - 6 - 4)t=4-10t$.

Step2: Find the parametric - form of a circle

The parametric equations of a circle in 2 - D with center $(a,b)$ and radius $r$ are $x=a + r\cos t$ and $y=b + r\sin t$. For a circle in 3 - D parallel to the $xy$ - plane with center $(2,4,-3)$ and radius $r = 5$, the parametric equations are $x = 2+5\cos t$, $y = 4+5\sin t$, $z=-3$. When the circle is counter - clockwise, $t\in[0,2\pi]$.

Step3: Find the parametric - form of an ellipse

The standard form of an ellipse is $\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1$. The parametric equations are $x = h + a\cos t$ and $y = k + b\sin t$.
For the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{25}=1$, we have $h = 0,k = 0,a = 6,b = 5$, so $x = 6\cos t$, $y = 5\sin t$.
For the ellipse $\frac{(x - 1)^2}{36}+\frac{(y + 2)^2}{25}=1$, we have $h = 1,k=-2,a = 6,b = 5$, so $x = 1+6\cos t$, $y=-2 + 5\sin t$.

Answer:

(a) The parametric equations of the line are $x = 3t$, $y = 2-3t$, $z = 4-10t$, $t\in R$.
(b) The parametric equations of the circle are $x = 2+5\cos t$, $y = 4+5\sin t$, $z=-3$, $t\in[0,2\pi]$.
(c) The parametric equations of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{25}=1$ are $x = 6\cos t$, $y = 5\sin t$, $t\in[0,2\pi]$. The parametric equations of the ellipse $\frac{(x - 1)^2}{36}+\frac{(y + 2)^2}{25}=1$ are $x = 1+6\cos t$, $y=-2 + 5\sin t$, $t\in[0,2\pi]$.
(d) The parametric equations of the ellipse $\frac{(x - 1)^2}{36}+\frac{(y + 2)^2}{25}=1$ are $x = 1+6\cos t$, $y=-2 + 5\sin t$, $t\in[0,2\pi]$.