We are given two points $(-2, 8)$ and $(-5, -8)$ that lie on a line, with the corresponding parameter values $t = 2$ and $t = 0$, respectively. We need to find the parametric equations of the line.

Step 1: General form of parametric equations

The parametric equations of a line are typically expressed as: $x = x_0 + at$ $y = y_0 + bt$ where $(x_0, y_0)$ is a point on the line, and $a$ and $b$ represent the direction ratios (change in $x$ and $y$ per unit change in $t$).

Step 2: Find the direction ratios $a$ and $b$

To find the direction ratios, we can calculate the change in $x$ and $y$ between the two given points.

For the $x$-coordinate: $\Delta x = x_1 - x_2 = -2 - (-5) = 3$ For the $y$-coordinate: $\Delta y = y_1 - y_2 = 8 - (-8) = 16$

Since the change in $t$ between these two points is $2 - 0 = 2$, the direction ratios $a$ and $b$ will be: $a = \frac{\Delta x}{\Delta t} = \frac{3}{2}, \quad b = \frac{\Delta y}{\Delta t} = \frac{16}{2} = 8$

Step 3: Write the parametric equations

Now, we can use the point $(-5, -8)$ at $t = 0$ as the initial point $(x_0, y_0)$, and the direction ratios $a = \frac{3}{2}, b = 8$.

The parametric equations of the line will be: $x(t) = -5 + \frac{3}{2}t$ $y(t) = -8 + 8t$

Thus, the parametric equations of the line are: $x(t) = -5 + \frac{3}{2}t, \quad y(t) = -8 + 8t$

Would you like more details on any part of the process? Here are some related questions:

1. How do we convert parametric equations to a Cartesian form?
2. What are the direction ratios, and how do they relate to the slope of the line?
3. Can you explain how to graph parametric equations step by step?
4. What happens if the parameter $t$ corresponds to different values for other points?
5. How can we derive parametric equations for curves other than lines?

Tip:

When dealing with parametric equations, it's often helpful to find the direction ratios, as they determine how fast each variable changes with respect to the parameter.