Math Problem Statement
Sean {x = 2 - t, y = 1 + 4t, t ∈ R}, las ecuaciones paramétricas de una recta r. a) ¿Los puntos P(1,5) y Q(3,-2) pertenecen a r? b) ¿Para qué valor del parámetro t se obtiene el punto (-2,17)? c) ¿Para qué valores de t se obtienen los puntos del segmento determinado por las intersecciones de la recta con los ejes coordenados? d) Calcule el área del triángulo que forma la recta con los ejes coordenados. e) Escriba otras ecuaciones paramétricas de la misma recta.
Solution
The problem provides parametric equations for a line in the form:
x = 2 - t \\ y = 1 + 4t \\ t \in \mathbb{R} \end{cases}$$ Let's break down and address each question: ### a) Do the points $$P(1, 5)$$ and $$Q(3, -2)$$ belong to $$r$$? To check if these points belong to the line, we need to find whether there exists a value of $$t$$ such that the parametric equations yield these points. 1. **Point $$P(1, 5)$$:** - From the equation $$x = 2 - t$$, set $$x = 1$$: $$1 = 2 - t \quad \Rightarrow \quad t = 1.$$ - Check if $$y = 5$$ holds when $$t = 1$$: $$y = 1 + 4(1) = 5.$$ So, $$P(1, 5)$$ belongs to the line $$r$$. 2. **Point $$Q(3, -2)$$:** - From the equation $$x = 2 - t$$, set $$x = 3$$: $$3 = 2 - t \quad \Rightarrow \quad t = -1.$$ - Check if $$y = -2$$ holds when $$t = -1$$: $$y = 1 + 4(-1) = 1 - 4 = -3.$$ Since $$y \neq -2$$, the point $$Q(3, -2)$$ does **not** belong to the line $$r$$. ### b) For what value of $$t$$ is the point $$(-2, 17)$$ on the line? We solve forAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Coordinate Geometry
Intersection with Axes
Area of Triangle
Formulas
x = 2 - t, y = 1 + 4t
Intersection with x-axis: y = 0
Intersection with y-axis: x = 0
Area of triangle = 1/2 * base * height
Theorems
Equations of a Line
Intersection Theorems for Coordinate Axes
Suitable Grade Level
Grades 10-12
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