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Consider the following statement. The parametric equations x = X1 + (x2 - X1)t, y = Y1 + (Y2 – Y1)t where 0 st s 1, describe the line segment that joins the points P,(x1, y1) and P2(x2, Y2). The following is a proposed proof for the statement. 1. When t = 0 the curve passes through P, (x,, y,) and when t = 1 the curve passes through P2(x2, Y2). 2. For 0

Consider the following statement. The parametric equations x = X1 + (x2 - X1)t, y = Y1 + (Y2 – Y1)t where 0 st s 1, describe the line segment that joins the points P,(x1, y1) and P2(x2, Y2). The following is a proposed proof for the statement. 1. When t = 0 the curve passes through P, (x,, y,) and when t = 1 the curve passes through P2(x2, Y2). 2. For 0

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 1ECP: Sketch and describe the orientation of the curve given by the parametric equations...
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Consider the following statement. The parametric equations x = X1 + (X2 - X1)t, y = Y1 + (Y2 - Y1)t where 0 <t 1, describe the line segment that joins the points P, (x1,Y;) and P2(x2, Y2). The following is a proposed proof for the statement. 1. When t = 0 the curve passes through P,(x,, y,) and when t = 1 the curve passes through P2(x2, Y2). 2. For 0 <t < 1, x is strictly between y, and y2. and X2 and y is strictly between Y1 3. For every value of t, x and y satisfy the relation 72 71(x – x,), which is the equation of the line through X2 - X1 y – Y1 = P;(X1, Y1) and P2(x2, Y2). У - У1 x – X1 X - 4. Finally, any point (x, y) on that line will satisfy If %D Y2 - Y1 X2 - X1 y - Y1 x - X1 we let t = then substituting t into the given Y2 - Y1 X2 - X1 parametric equations yields the point (x, y), and any (x, y) on the line segment from P;(x1, Y1) to P2(x2, Y2) yields a value of t in [-1, 1]. 5. So the given parametric equations exactly specify the line segment from P, (x1, Y1) to P2(x2, Y2). Identify the error(s) in the proposed proof. (Select all that apply.) O The first sentence assumes t O and t 1, which cannot be assumed based on the given domain. The second sentence should say 0 <t < 1 instead of 0 < t < 1. Y2-Y1(x – x,) is a linear equation when this is not the X2 - X1 The third sentence claims that y - y1 case. The fourth sentence should say t in [0, 1] instead of t in [-1, 1]. The fifth sentence should say P(x, y) instead of P,(x2, y2).
Transcribed Image Text:Consider the following statement. The parametric equations x = X1 + (X2 - X1)t, y = Y1 + (Y2 - Y1)t where 0 <t 1, describe the line segment that joins the points P, (x1,Y;) and P2(x2, Y2). The following is a proposed proof for the statement. 1. When t = 0 the curve passes through P,(x,, y,) and when t = 1 the curve passes through P2(x2, Y2). 2. For 0 <t < 1, x is strictly between y, and y2. and X2 and y is strictly between Y1 3. For every value of t, x and y satisfy the relation 72 71(x – x,), which is the equation of the line through X2 - X1 y – Y1 = P;(X1, Y1) and P2(x2, Y2). У - У1 x – X1 X - 4. Finally, any point (x, y) on that line will satisfy If %D Y2 - Y1 X2 - X1 y - Y1 x - X1 we let t = then substituting t into the given Y2 - Y1 X2 - X1 parametric equations yields the point (x, y), and any (x, y) on the line segment from P;(x1, Y1) to P2(x2, Y2) yields a value of t in [-1, 1]. 5. So the given parametric equations exactly specify the line segment from P, (x1, Y1) to P2(x2, Y2). Identify the error(s) in the proposed proof. (Select all that apply.) O The first sentence assumes t O and t 1, which cannot be assumed based on the given domain. The second sentence should say 0 <t < 1 instead of 0 < t < 1. Y2-Y1(x – x,) is a linear equation when this is not the X2 - X1 The third sentence claims that y - y1 case. The fourth sentence should say t in [0, 1] instead of t in [-1, 1]. The fifth sentence should say P(x, y) instead of P,(x2, y2).
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