Worksheet: Equation of a Straight Line: Parametric Form

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In this worksheet, we will practice finding the equation of a straight line in parametric form using a point on the line and the vector direction of the line.

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

Q2:

Find the parametric equations of the straight line that makes an angle of 135∘ with the positive π‘₯-axis and passes through the point (1,βˆ’15).

  • A π‘₯ = 1 + 𝐾 , 𝑦 = 1 βˆ’ 1 5 𝐾
  • B π‘₯ = βˆ’ 1 5 βˆ’ 𝐾 , 𝑦 = 1 + 𝐾
  • C π‘₯ = 1 , 𝑦 = βˆ’ 1 5 βˆ’ 𝐾
  • D π‘₯ = 1 + 𝐾 , 𝑦 = βˆ’ 1 5 βˆ’ 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
  • C π‘Ÿ
  • D π‘Ÿ 2 

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

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

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

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

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

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

Q4:

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

  • A π‘₯ = 2 + π‘Ÿ 2 , 𝑦 = 3 βˆ’ π‘Ÿ √ 3 2
  • B π‘₯ = 2 βˆ’ π‘Ÿ 2 , 𝑦 = 3 + π‘Ÿ √ 3 2
  • C π‘₯ = 2 βˆ’ π‘Ÿ 2 , 𝑦 = 3 βˆ’ π‘Ÿ √ 3 2
  • D π‘₯ = 2 + π‘Ÿ √ 3 2 , 𝑦 = 3 + π‘Ÿ 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 c o s
  • B π‘Ÿ = π‘₯ + π‘Ž ( πœƒ ) = 𝑦 + 𝑏 ( πœƒ ) c o s s i n
  • C π‘Ÿ = π‘₯ βˆ’ π‘Ž ( πœƒ ) = 𝑦 βˆ’ 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‘₯=2𝑑+1,𝑦=βˆ’3𝑑+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 = 3 𝑑 + 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 = βˆ’ 3 𝑑 βˆ’ 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‘₯ = 3 𝑑 βˆ’ 2 , 𝑦 = βˆ’ 𝑑 βˆ’ 3
  • B π‘₯ = 2 𝑑 + 3 , 𝑦 = 3 𝑑 + 1
  • C π‘₯ = βˆ’ 2 𝑑 βˆ’ 3 , 𝑦 = βˆ’ 3 𝑑 + 1
  • D π‘₯ = 2 𝑑 + 3 , 𝑦 = βˆ’ 3 𝑑 βˆ’ 1
  • E π‘₯ = βˆ’ 2 𝑑 + 3 , 𝑦 = 3 𝑑 βˆ’ 1

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

  • A π‘₯ = 𝑑 βˆ’ 1 , 𝑦 = βˆ’ 3 2 𝑑 βˆ’ 2
  • B π‘₯ = 𝑑 βˆ’ 1 , 𝑦 = βˆ’ 3 2 𝑑 + 2
  • C π‘₯ = 𝑑 + 1 , 𝑦 = βˆ’ 3 2 𝑑 + 2
  • D π‘₯ = 4 𝑑 + 1 , 𝑦 = βˆ’ 6 𝑑 + 2
  • E π‘₯ = 4 𝑑 βˆ’ 1 , 𝑦 = βˆ’ 6 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 twice as fast as over 0≀𝑑≀1.
  • BOver 0≀𝑑≀2, the particle is moving one-third as fast as over 0≀𝑑≀1.
  • COver 0≀𝑑≀2, the particle is moving half as fast as over 0≀𝑑≀1.
  • DOver 0≀𝑑≀2, the particle is moving three times 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 𝐴𝐡 over the interval 1≀𝑑≀3.

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

Give the parameterization of 𝐡𝐢 over the interval 3≀𝑑≀5.

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

Q8:

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

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

Q9:

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

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

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 ⎩ 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
  • 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 ⎩ 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
  • E 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

Q11:

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

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

Q12:

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

  • AFalse
  • BTrue

Q13:

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

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

Q14:

The lines π‘₯=3π‘‘βˆ’2, 𝑦=3𝑑+2, 𝑧=9π‘‘βˆ’2 and π‘₯=π‘Žπ‘‘βˆ’2, 𝑦=𝑑+1, 𝑧=π‘π‘‘βˆ’2 are parallel. What is π‘Ž+𝑏?

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