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Find the points on the line nhere the ta x(t)%3D t3 - 3t +2, Y (t) =3-3t?+2

Find the points on the line nhere the ta x(t)%3D t3 - 3t +2, Y (t) =3-3t?+2

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.6: Parametric Equations
Problem 5ECP: Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle...
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Find the points of the line where the tangent is vertical 

## Problem Statement Find the points on the line where the two curves meet. ### Given Parametric Equations: \[ x(t) = t^3 - 3t + 2 \] \[ y(t) = t^3 - 3t^2 + 2 \] ### Instructions: 1. **Identify the Intersection Points:** - Determine the values of \( t \) where the two curves intersect. - Substitute these values back into the parametric equations to find the coordinates of the intersection points. 2. **Graphical Representation:** - If applicable, graph the parametric equations to visualize the points of intersection. ### Explanation: To locate the intersection points, you need to solve the equations simultaneously for the parameter \( t \). This will involve setting the \( x(t) \) equation equal to the \( y(t) \) equation and solving for \( t \). #### Step-by-Step Solution: 1. **Set the two equations equal to each other:** \[ t^3 - 3t + 2 = t^3 - 3t^2 + 2 \] 2. **Simplify the equation:** \[ -3t + 2 = -3t^2 + 2 \] \[ -3t^2 + 3t = 0 \] \[ 3t(t - 1) = 0 \] 3. **Solve for \( t \):** \[ t = 0 \] or \[ t = 1 \] 4. **Substitute \( t \) back into the parametric equations to find the coordinates:** - For \( t = 0 \): \[ x(0) = 0^3 - 3(0) + 2 = 2 \] \[ y(0) = 0^3 - 3(0)^2 + 2 = 2 \] Intersection point: \((2, 2)\) - For \( t = 1 \): \[ x(1) = 1^3 - 3(1) + 2 = 0 \] \[ y(1) = 1^3 - 3(1)^2 + 2 = 0 \] Intersection point: \((0, 0)\) ### Conclusion: The points of intersection are
Transcribed Image Text:## Problem Statement Find the points on the line where the two curves meet. ### Given Parametric Equations: \[ x(t) = t^3 - 3t + 2 \] \[ y(t) = t^3 - 3t^2 + 2 \] ### Instructions: 1. **Identify the Intersection Points:** - Determine the values of \( t \) where the two curves intersect. - Substitute these values back into the parametric equations to find the coordinates of the intersection points. 2. **Graphical Representation:** - If applicable, graph the parametric equations to visualize the points of intersection. ### Explanation: To locate the intersection points, you need to solve the equations simultaneously for the parameter \( t \). This will involve setting the \( x(t) \) equation equal to the \( y(t) \) equation and solving for \( t \). #### Step-by-Step Solution: 1. **Set the two equations equal to each other:** \[ t^3 - 3t + 2 = t^3 - 3t^2 + 2 \] 2. **Simplify the equation:** \[ -3t + 2 = -3t^2 + 2 \] \[ -3t^2 + 3t = 0 \] \[ 3t(t - 1) = 0 \] 3. **Solve for \( t \):** \[ t = 0 \] or \[ t = 1 \] 4. **Substitute \( t \) back into the parametric equations to find the coordinates:** - For \( t = 0 \): \[ x(0) = 0^3 - 3(0) + 2 = 2 \] \[ y(0) = 0^3 - 3(0)^2 + 2 = 2 \] Intersection point: \((2, 2)\) - For \( t = 1 \): \[ x(1) = 1^3 - 3(1) + 2 = 0 \] \[ y(1) = 1^3 - 3(1)^2 + 2 = 0 \] Intersection point: \((0, 0)\) ### Conclusion: The points of intersection are
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