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Lesson: Equation of a Straight Line in Parametric Form

In this lesson, we will learn how to find the equation of a straight line in parametric form using a point on the line and the vector direction of the line.

Sample Question Videos

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Worksheet • 14 Questions • 2 Videos

Q1:

Find the parametric equations of the straight line that passes through the point ( βˆ’ 9 , 8 ) with direction vector ( 4 , βˆ’ 7 ) .

  • A π‘₯ = βˆ’ 9 + 4 𝐾 , 𝑦 = 8 βˆ’ 7 𝐾
  • B π‘₯ = 8 βˆ’ 7 𝐾 , 𝑦 = βˆ’ 9 + 4 𝐾
  • C π‘₯ = 8 + 4 𝐾 , 𝑦 = βˆ’ 9 βˆ’ 7 𝐾
  • D π‘₯ = βˆ’ 9 + 8 𝐾 , 𝑦 = 4 βˆ’ 7 𝐾

Q2:

Find the parametric equations of the straight line that makes an angle of 1 3 5 ∘ with the positive π‘₯ -axis and passes through the point ( 1 , βˆ’ 1 5 ) .

  • A π‘₯ = 1 + 𝐾 , 𝑦 = βˆ’ 1 5 βˆ’ 𝐾
  • B π‘₯ = 1 + 𝐾 , 𝑦 = 1 βˆ’ 1 5 𝐾
  • C π‘₯ = 1 , 𝑦 = βˆ’ 1 5 βˆ’ 𝐾
  • D π‘₯ = βˆ’ 1 5 βˆ’ 𝐾 , 𝑦 = 1 + 𝐾

Q3:

Consider the line shown that passes through the point ( 3 , 4 ) and makes an angle of 45 degrees with the positive π‘₯ -axis.

Suppose that the distance between ( 3 , 4 ) and any point ( π‘₯ , 𝑦 ) on the line is π‘Ÿ .

Write, in terms of π‘Ÿ , an expression for the horizontal distance π‘₯ βˆ’ 3 between the two points.

  • A π‘Ÿ √ 2
  • B π‘Ÿ 2 2
  • C π‘Ÿ 2
  • D π‘Ÿ

Write, in terms of π‘Ÿ , an expression for the vertical distance 𝑦 βˆ’ 4 between the two points.

  • A π‘Ÿ √ 2
  • B π‘Ÿ 2 2
  • C π‘Ÿ 2
  • D π‘Ÿ

Hence, write a pair of parametric equations which describe the line.

  • A π‘₯ = 3 + π‘Ÿ √ 2 , 𝑦 = 4 + π‘Ÿ √ 2
  • B π‘₯ = 3 βˆ’ π‘Ÿ √ 2 , 𝑦 = 4 + π‘Ÿ √ 2
  • C π‘₯ = 3 βˆ’ π‘Ÿ √ 2 , 𝑦 = 4 βˆ’ π‘Ÿ √ 2
  • D π‘₯ = 3 + π‘Ÿ √ 2 , 𝑦 = 4 βˆ’ π‘Ÿ √ 2

Find the coordinates of the point on the line which is at a distance of 4 from ( 3 , 4 ) .

  • A ( 3 + 2 √ 2 , 4 + 2 √ 2 )
  • B ( 3 βˆ’ 2 √ 2 , 4 + 2 √ 2 )
  • C ( 3 βˆ’ 2 √ 2 , 4 βˆ’ 2 √ 2 )
  • D ( 3 + 2 √ 2 , 4 βˆ’ 2 √ 2 )

Q4:

Write a pair of parametric equations with parameter π‘Ÿ describing the shown line.

  • A π‘₯ = 2 + π‘Ÿ √ 3 2 , 𝑦 = 3 + π‘Ÿ 2
  • B π‘₯ = 2 βˆ’ π‘Ÿ 2 , 𝑦 = 3 βˆ’ π‘Ÿ √ 3 2
  • C π‘₯ = 2 βˆ’ π‘Ÿ 2 , 𝑦 = 3 + π‘Ÿ √ 3 2
  • D π‘₯ = 2 + π‘Ÿ 2 , 𝑦 = 3 βˆ’ π‘Ÿ √ 3 2

Q5:

Write the parametric equation of the straight line that passes through the point ( π‘Ž , 𝑏 ) and makes an angle of πœƒ with the positive π‘₯ -axis as shown.

  • A π‘Ÿ = π‘₯ βˆ’ π‘Ž ( πœƒ ) = 𝑦 βˆ’ 𝑏 ( πœƒ ) c o s s i n
  • B π‘Ÿ = π‘₯ βˆ’ π‘Ž ( πœƒ ) = 𝑦 βˆ’ 𝑏 ( πœƒ ) c o s c o s
  • C π‘Ÿ = π‘₯ + π‘Ž ( πœƒ ) = 𝑦 + 𝑏 ( πœƒ ) c o s s i n
  • D π‘Ÿ = π‘₯ βˆ’ π‘Ž ( πœƒ ) = 𝑦 βˆ’ 𝑏 ( πœƒ ) s i n s i n

Q6:

The equations π‘₯ = 2 𝑑 + 1 , 𝑦 = βˆ’ 3 𝑑 + 2 parameterize the line segment between ( 1 , 2 ) and ( 3 , βˆ’ 1 ) over the interval 0 ≀ 𝑑 ≀ 1 .

Which of the following is a parameterization of the line segment on βˆ’ 1 ≀ 𝑑 ≀ 0 ?

  • A π‘₯ = 2 𝑑 + 3 , 𝑦 = βˆ’ 3 𝑑 βˆ’ 1
  • B π‘₯ = 3 𝑑 βˆ’ 2 , 𝑦 = βˆ’ 𝑑 βˆ’ 3
  • C π‘₯ = βˆ’ 2 𝑑 + 3 , 𝑦 = 3 𝑑 βˆ’ 1
  • D π‘₯ = 2 𝑑 + 3 , 𝑦 = 3 𝑑 + 1
  • E π‘₯ = βˆ’ 2 𝑑 βˆ’ 3 , 𝑦 = βˆ’ 3 𝑑 + 1

Which of the following is a parameterization of the line segment on 0 ≀ 𝑑 ≀ 1 that starts at ( 3 , βˆ’ 1 ) and ends at ( 1 , 2 ) ?

  • A π‘₯ = βˆ’ 2 𝑑 + 3 , 𝑦 = 3 𝑑 βˆ’ 1
  • B π‘₯ = 2 𝑑 + 3 , 𝑦 = 3 𝑑 + 1
  • C π‘₯ = 2 𝑑 + 3 , 𝑦 = βˆ’ 3 𝑑 βˆ’ 1
  • D π‘₯ = 3 𝑑 βˆ’ 2 , 𝑦 = βˆ’ 𝑑 βˆ’ 3
  • E π‘₯ = βˆ’ 2 𝑑 βˆ’ 3 , 𝑦 = βˆ’ 3 𝑑 + 1

Which of the following is a parameterization of the line segment on 0 ≀ 𝑑 ≀ 2 ?

  • A π‘₯ = 𝑑 + 1 , 𝑦 = βˆ’ 3 2 𝑑 + 2
  • B π‘₯ = 𝑑 βˆ’ 1 , 𝑦 = βˆ’ 3 2 𝑑 + 2
  • C π‘₯ = 4 𝑑 + 1 , 𝑦 = βˆ’ 6 𝑑 + 2
  • D π‘₯ = 4 𝑑 βˆ’ 1 , 𝑦 = βˆ’ 6 𝑑 βˆ’ 2
  • E π‘₯ = 𝑑 βˆ’ 1 , 𝑦 = βˆ’ 3 2 𝑑 βˆ’ 2

If the parameterizations you have given above correspond to a particle moving along the line segment, how does the parameterization over interval 0 ≀ 𝑑 ≀ 2 relate to the one over 0 ≀ 𝑑 ≀ 1 ?

  • AOver 0 ≀ 𝑑 ≀ 2 , the particle is moving half as fast as over 0 ≀ 𝑑 ≀ 1 .
  • BOver 0 ≀ 𝑑 ≀ 2 , the particle is moving twice as fast as over 0 ≀ 𝑑 ≀ 1 .
  • COver 0 ≀ 𝑑 ≀ 2 , the particle is moving three times as fast as over 0 ≀ 𝑑 ≀ 1 .
  • DOver 0 ≀ 𝑑 ≀ 2 , the particle is moving one-third as fast as over 0 ≀ 𝑑 ≀ 1 .

Q7:

Consider the points 𝐴 , 𝐡 , and 𝐢 and the line segments in the figure.

Give the parameterization of 𝐴 𝐡 over the interval 1 ≀ 𝑑 ≀ 3 .

  • A π‘₯ = 1 , 𝑦 = 𝑑
  • B π‘₯ = 1 , 𝑦 = 2 𝑑 βˆ’ 1
  • C π‘₯ = 𝑑 , 𝑦 = 1
  • D π‘₯ = 𝑑 , 𝑦 = 0
  • E π‘₯ = 0 , 𝑦 = 𝑑

Give the parameterization of 𝐡 𝐢 over the interval 3 ≀ 𝑑 ≀ 5 .

  • A π‘₯ = 𝑑 βˆ’ 2 , 𝑦 = 3
  • B π‘₯ = 2 , 𝑦 = 3
  • C π‘₯ = 𝑑 βˆ’ 3 , 𝑦 = 2
  • D π‘₯ = 3 , 𝑦 = 𝑑 βˆ’ 2
  • E π‘₯ = 2 , 𝑦 = 𝑑 βˆ’ 3

Find functions 𝑓 and 𝑔 defined for 1 ≀ 𝑑 ≀ 5 so that π‘₯ = 𝑓 ( 𝑑 ) , 𝑦 = 𝑔 ( 𝑑 ) parameterizes the given path from 𝐴 to 𝐢 .

  • A 𝑓 ( 𝑑 ) =  1 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 2 3 < 𝑑 ≀ 5 i f i f , 𝑔 ( 𝑑 ) =  𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 i f i f
  • B 𝑓 ( 𝑑 ) =  𝑑 1 ≀ 𝑑 ≀ 3 , 2 3 < 𝑑 ≀ 5 i f i f , 𝑔 ( 𝑑 ) =  1 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 3 3 < 𝑑 ≀ 5 i f i f
  • C 𝑓 ( 𝑑 ) =  𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 i f i f , 𝑔 ( 𝑑 ) =  1 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 2 3 < 𝑑 ≀ 5 i f i f
  • D 𝑓 ( 𝑑 ) =  1 1 ≀ 𝑑 ≀ 3 , 𝑑 + 2 3 < 𝑑 ≀ 5 i f i f , 𝑔 ( 𝑑 ) =  𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 i f i f
  • E 𝑓 ( 𝑑 ) =  1 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 3 3 < 𝑑 ≀ 5 i f i f , 𝑔 ( 𝑑 ) =  𝑑 1 ≀ 𝑑 ≀ 3 , 2 3 < 𝑑 ≀ 5 i f i f

Q8:

Let 𝐴 = ( 1 , 1 ) and 𝐡 = ( 1 , 3 ) . Which of the following is a parameterization of 𝐴 𝐡 over 0 ≀ 𝑑 ≀ 1 that starts at 𝐴 and ends at 𝐡 .

  • A π‘₯ = 1 , 𝑦 = 2 𝑑 + 1
  • B π‘₯ = 2 𝑑 + 1 , 𝑦 = 1
  • C π‘₯ = 1 , 𝑦 = 𝑑 + 1
  • D π‘₯ = 𝑑 + 1 , 𝑦 = 1
  • E π‘₯ = 1 , 𝑦 = 2 ( 𝑑 + 1 )

Q9:

Let 𝐴 = ( 1 , 1 ) and 𝐡 = ( 1 , 2 ) . Which of the following is a parameterization of 𝐴 𝐡 over 0 ≀ 𝑑 ≀ 1 that starts at 𝐡 and ends at 𝐴 .

  • A π‘₯ = 1 , 𝑦 = 2 βˆ’ 𝑑
  • B π‘₯ = 𝑑 + 1 , 𝑦 = 1
  • C π‘₯ = 1 , 𝑦 = 𝑑 + 1
  • D π‘₯ = 2 βˆ’ 𝑑 , 𝑦 = 1
  • E π‘₯ = 1 , 𝑦 = 2 + 𝑑

Q10:

Find the parameterization π‘₯ = 𝑓 ( 𝑑 ) , 𝑦 = 𝑔 ( 𝑑 ) of the path 𝐴 , 𝐡 , 𝐢 , 𝐷 using the interval 1 ≀ 𝑑 ≀ 9 .

  • A 𝑓 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 1 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 2 3 < 𝑑 ≀ 5 , 3 5 < 𝑑 ≀ 7 , 1 0 βˆ’ 𝑑 7 < 𝑑 ≀ 9 i f i f i f i f , 𝑔 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 , 8 βˆ’ 𝑑 5 < 𝑑 ≀ 7 , 1 7 < 𝑑 ≀ 9 i f i f i f i f
  • B 𝑓 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 3 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 2 3 < 𝑑 ≀ 5 , 1 5 < 𝑑 ≀ 7 , 1 0 βˆ’ 𝑑 7 < 𝑑 ≀ 9 i f i f i f i f , 𝑔 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 , 8 βˆ’ 𝑑 5 < 𝑑 ≀ 7 , 1 7 < 𝑑 ≀ 9 i f i f i f i f
  • C 𝑓 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 1 1 ≀ 𝑑 ≀ 3 , 𝑑 βˆ’ 2 3 < 𝑑 ≀ 5 , 3 5 < 𝑑 ≀ 7 , 𝑑 βˆ’ 1 0 7 < 𝑑 ≀ 9 i f i f i f i f , 𝑔 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 , 𝑑 βˆ’ 8 5 < 𝑑 ≀ 7 , 1 7 < 𝑑 ≀ 9 i f i f i f i f
  • D 𝑓 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 1 1 ≀ 𝑑 ≀ 3 , 𝑑 + 2 3 < 𝑑 ≀ 5 , 3 5 < 𝑑 ≀ 7 , 𝑑 + 1 0 7 < 𝑑 ≀ 9 i f i f i f i f , 𝑔 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 , 𝑑 βˆ’ 8 5 < 𝑑 ≀ 7 , 1 7 < 𝑑 ≀ 9 i f i f i f i f
  • E 𝑓 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 1 1 ≀ 𝑑 ≀ 3 , 𝑑 + 2 3 < 𝑑 ≀ 5 , 3 5 < 𝑑 ≀ 7 , 1 0 + 𝑑 7 < 𝑑 ≀ 9 i f i f i f i f , 𝑔 ( 𝑑 ) = ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝑑 1 ≀ 𝑑 ≀ 3 , 3 3 < 𝑑 ≀ 5 , 8 βˆ’ 𝑑 5 < 𝑑 ≀ 7 , 1 7 < 𝑑 ≀ 9 i f i f i f i f

Q11:

Let 𝐴 = ( 1 , 1 ) and 𝐡 = ( 1 , 2 ) . Find the parameterization of 𝐴 𝐡 over 0 ≀ 𝑑 ≀ 1 that starts at 𝐴 and ends at 𝐡 .

  • A π‘₯ = 1 , 𝑦 = 𝑑 + 1
  • B π‘₯ = 𝑑 + 1 , 𝑦 = 𝑑
  • C π‘₯ = 1 , 𝑦 = 𝑑
  • D π‘₯ = 𝑑 + 1 , 𝑦 = 𝑑
  • E π‘₯ = 1 , 𝑦 = 𝑑 βˆ’ 1

Q12:

A cube with side 3 sits with a vertex at the origin and three sides along the positive axes. Find the parametric equations of the main diagonal from the origin.

  • A π‘₯ = 3 𝑑 , 𝑦 = 3 𝑑 , 𝑧 = 3 𝑑
  • B π‘₯ = 3 , 𝑦 = 3 , 𝑧 = 3
  • C π‘₯ = 1 + 3 𝑑 , 𝑦 = 1 + 3 𝑑 , 𝑧 = 1 + 3 𝑑
  • D π‘₯ = 3 + 𝑑 , 𝑦 = 3 + 𝑑 , 𝑧 = 3 + 𝑑

Q13:

True or False: There is only one way to parameterize the line segment from ( 1 , 2 ) to ( 3 , βˆ’ 1 ) .

  • Atrue
  • Bfalse

Q14:

Find the parametric equations of the straight line that passes through the point ( 9 , βˆ’ 7 ) with direction vector ( 3 , 2 ) .

  • A π‘₯ = 9 + 3 𝐾 , 𝑦 = βˆ’ 7 + 2 𝐾
  • B π‘₯ = βˆ’ 7 + 2 𝐾 , 𝑦 = 9 + 3 𝐾
  • C π‘₯ = βˆ’ 7 + 3 𝐾 , 𝑦 = 9 + 2 𝐾
  • D π‘₯ = 9 βˆ’ 7 𝐾 , 𝑦 = 3 + 2 𝐾
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