Question

Eliminate the parameter and obtain the standard form of the rectangular equation. line through (x1, y1) and (x2, y2): x = x1 + t(x2 − x1), y = y1 + t(y2 − y1) use your result to find a set of parametric equations for the line or conic. (enter your answers as a comma-separated list.) line: passes through (0, 0) and (4, -4)

Answer 1

The parametric equations for the line passes through the points (X₁ , Y₁) and (X₂ , Y₂) are :

X = X₁ + t (X₂ - X₁) ......................... (1)

Y = Y₁ + t (Y₂ - Y₁) .......................... (2)

For eliminating the parameter (t) , first we will solve any one equation for 't' and then substitute that into another equation.

From the equation (1),

⇒ t =
`(X- X1)/((X2 - X1))`

Substituting this t =
`(X- X1)/((X2 - X1))` into the equation (2), we will get :

Y = Y₁ +
`((X- X1))/((X2 - X1))` (Y₂ - Y₁)

Y = Y₁ +
`((X- X1)(Y2 - Y1))/((X2 - X1))`

So, this is the standard form of rectangular equation.

For two given points (0, 0) and (4, -4)

X₁ = 0 , Y₁ = 0, X₂ = 4 and Y₂ = -4

For finding the parametric equations, we will plug these values into the given parametric equations.

X = X₁ + t (X₂ - X₁)

⇒ X = 0+ t (4 - 0)

⇒ X = 4t

and Y = Y₁ + t (Y₂ - Y₁)

⇒ Y = 0+ t (-4 - 0)

⇒ Y = - 4t

So, the parametric equations for the line passing through (0,0) and (4, -4) are: X = 4t and Y = -4t