Question

# Find parametric equations for the curve with the given properties. The line with slope -2, passing through (-10,-20)

Equations
Find parametric equations for the curve with the given properties.
The line with slope -2, passing through (-10,-20)

2021-03-31
To find the parametric equation of the line with slope -2 passing through (-10,-20)
Given slope $$\displaystyle{m}={\frac{{\triangle{y}}}{{\triangle{x}}}}={\frac{{{y}_{{{s}}}}}{{{x}_{{{s}}}}}}=-{2}$$
$$\displaystyle\Rightarrow{x}_{{{s}}}={1},{y}_{{{s}}}=-{2}$$
Also $$\displaystyle{\left({x}_{{{1}}},{y}_{{{1}}}\right)}={\left(-{10},-{20}\right)}$$
The parametric equations of the line is given by
$$\displaystyle{x}={x}_{{{1}}}+{x}_{{{s}}}\cdot{t}$$
$$\displaystyle{y}={y}_{{{1}}}+{y}_{{{s}}}\cdot{t}$$
Substituting the values we have
$$\displaystyle{x}={x}_{{{1}}}+{x}_{{{s}}}\cdot{t}=-{10}+{1}{t}={t}-{10}$$
$$\displaystyle{y}={y}_{{{1}}}+{y}_{{{s}}}\cdot{t}=--{20}-{2}{t}=-{2}{t}-{20}$$
Hence x=t-10, y=-2t-20 are the required parametric equations.