Worksheet: Equation of a Straight Line in 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 ( − 9 , 8 ) with direction vector ( 4 , − 7 ) .

  • A 𝑥 = − 9 + 8 𝐾 , 𝑦 = 4 − 7 𝐾
  • B 𝑥 = 8 − 7 𝐾 , 𝑦 = − 9 + 4 𝐾
  • C 𝑥 = 8 + 4 𝐾 , 𝑦 = − 9 − 7 𝐾
  • D 𝑥 = − 9 + 4 𝐾 , 𝑦 = 8 − 7 𝐾

Q2:

Find the parametric equations of the straight line that makes an angle of 1 3 5 ∘ with the positive 𝑥 -axis and passes through the point ( 1 , − 1 5 ) .

  • A 𝑥 = − 1 5 − 𝐾 , 𝑦 = 1 + 𝐾
  • B 𝑥 = 1 + 𝐾 , 𝑦 = 1 − 1 5 𝐾
  • C 𝑥 = 1 , 𝑦 = − 1 5 − 𝐾
  • D 𝑥 = 1 + 𝐾 , 𝑦 = − 1 5 − 𝐾

Q3:

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

Suppose that the distance between ( 3 , 4 ) and any point ( 𝑥 , 𝑦 ) on the line is 𝑟 .

Write, in terms of 𝑟 , an expression for the horizontal distance 𝑥 − 3 between the two points.

  • A 𝑟
  • B 𝑟 2 2
  • C 𝑟 2
  • D 𝑟 √ 2

Write, in terms of 𝑟 , an expression for the vertical distance 𝑦 − 4 between the two points.

  • A 𝑟 2
  • B 𝑟
  • C 𝑟 2 2
  • D 𝑟 √ 2

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

  • A 𝑥 = 3 + 𝑟 √ 2 , 𝑦 = 4 + 𝑟 √ 2
  • B 𝑥 = 3 − 𝑟 √ 2 , 𝑦 = 4 + 𝑟 √ 2
  • C 𝑥 = 3 + 𝑟 √ 2 , 𝑦 = 4 − 𝑟 √ 2
  • D 𝑥 = 3 − 𝑟 √ 2 , 𝑦 = 4 − 𝑟 √ 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 − 2 √ 2 , 4 − 2 √ 2 )
  • C ( 3 − 2 √ 2 , 4 + 2 √ 2 )
  • D ( 3 + 2 √ 2 , 4 − 2 √ 2 )

Q4:

Write a pair of parametric equations with parameter 𝑟 describing the shown line.

  • A 𝑥 = 2 + 𝑟 2 , 𝑦 = 3 − 𝑟 √ 3 2
  • B 𝑥 = 2 − 𝑟 2 , 𝑦 = 3 − 𝑟 √ 3 2
  • C 𝑥 = 2 − 𝑟 2 , 𝑦 = 3 + 𝑟 √ 3 2
  • D 𝑥 = 2 + 𝑟 √ 3 2 , 𝑦 = 3 + 𝑟 2

Q5:

Write the parametric equation of the straight line that passes through the point ( 𝑎 , 𝑏 ) and makes an angle of 𝜃 with the positive 𝑥 -axis as shown.

  • A 𝑟 = 𝑥 − 𝑎 ( 𝜃 ) = 𝑦 − 𝑏 ( 𝜃 ) s i n s i n
  • B 𝑟 = 𝑥 − 𝑎 ( 𝜃 ) = 𝑦 − 𝑏 ( 𝜃 ) c o s c o s
  • C 𝑟 = 𝑥 + 𝑎 ( 𝜃 ) = 𝑦 + 𝑏 ( 𝜃 ) c o s s i n
  • D 𝑟 = 𝑥 − 𝑎 ( 𝜃 ) = 𝑦 − 𝑏 ( 𝜃 ) c o s s i n

Q6:

The equations 𝑥 = 2 𝑡 + 1 , 𝑦 = − 3 𝑡 + 2 parameterize the line segment between ( 1 , 2 ) and ( 3 , − 1 ) over the interval 0 ≤ 𝑡 ≤ 1 .

Which of the following is a parameterization of the line segment on − 1 ≤ 𝑡 ≤ 0 ?

  • A 𝑥 = − 2 𝑡 − 3 , 𝑦 = − 3 𝑡 + 1
  • B 𝑥 = − 2 𝑡 + 3 , 𝑦 = 3 𝑡 − 1
  • C 𝑥 = 2 𝑡 + 3 , 𝑦 = 3 𝑡 + 1
  • D 𝑥 = 2 𝑡 + 3 , 𝑦 = − 3 𝑡 − 1
  • E 𝑥 = 3 𝑡 − 2 , 𝑦 = − 𝑡 − 3

Which of the following is a parameterization of the line segment on 0 ≤ 𝑡 ≤ 1 that starts at ( 3 , − 1 ) and ends at ( 1 , 2 ) ?

  • A 𝑥 = − 2 𝑡 + 3 , 𝑦 = 3 𝑡 − 1
  • B 𝑥 = 3 𝑡 − 2 , 𝑦 = − 𝑡 − 3
  • C 𝑥 = − 2 𝑡 − 3 , 𝑦 = − 3 𝑡 + 1
  • D 𝑥 = 2 𝑡 + 3 , 𝑦 = − 3 𝑡 − 1
  • E 𝑥 = 2 𝑡 + 3 , 𝑦 = 3 𝑡 + 1

Which of the following is a parameterization of the line segment on 0 ≤ 𝑡 ≤ 2 ?

  • A 𝑥 = 𝑡 − 1 , 𝑦 = − 3 2 𝑡 + 2
  • B 𝑥 = 4 𝑡 + 1 , 𝑦 = − 6 𝑡 + 2
  • C 𝑥 = 𝑡 + 1 , 𝑦 = − 3 2 𝑡 + 2
  • D 𝑥 = 𝑡 − 1 , 𝑦 = − 3 2 𝑡 − 2
  • E 𝑥 = 4 𝑡 − 1 , 𝑦 = − 6 𝑡 − 2

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

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

Q7:

Consider the points 𝐴 , 𝐵 , and 𝐶 and the line segments in the figure.

Give the parameterization of 𝐴 𝐵 over the interval 1 ≤ 𝑡 ≤ 3 .

  • A 𝑥 = 0 , 𝑦 = 𝑡
  • B 𝑥 = 𝑡 , 𝑦 = 1
  • C 𝑥 = 𝑡 , 𝑦 = 0
  • D 𝑥 = 1 , 𝑦 = 𝑡
  • E 𝑥 = 1 , 𝑦 = 2 𝑡 − 1

Give the parameterization of 𝐵 𝐶 over the interval 3 ≤ 𝑡 ≤ 5 .

  • A 𝑥 = 𝑡 − 2 , 𝑦 = 3
  • B 𝑥 = 3 , 𝑦 = 𝑡 − 2
  • C 𝑥 = 2 , 𝑦 = 𝑡 − 3
  • D 𝑥 = 𝑡 − 3 , 𝑦 = 2
  • E 𝑥 = 2 , 𝑦 = 3

Find functions 𝑓 and 𝑔 defined for 1 ≤ 𝑡 ≤ 5 so that 𝑥 = 𝑓 ( 𝑡 ) , 𝑦 = 𝑔 ( 𝑡 ) parameterizes the given path from 𝐴 to 𝐶 .

  • A 𝑓 ( 𝑡 ) = 𝑡 1 ≤ 𝑡 ≤ 3 , 2 3 < 𝑡 ≤ 5 i f i f , 𝑔 ( 𝑡 ) = 1 1 ≤ 𝑡 ≤ 3 , 𝑡 − 3 3 < 𝑡 ≤ 5 i f i f
  • B 𝑓 ( 𝑡 ) = 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 i f i f , 𝑔 ( 𝑡 ) = 1 1 ≤ 𝑡 ≤ 3 , 𝑡 − 2 3 < 𝑡 ≤ 5 i f i f
  • C 𝑓 ( 𝑡 ) = 1 1 ≤ 𝑡 ≤ 3 , 𝑡 − 2 3 < 𝑡 ≤ 5 i f i f , 𝑔 ( 𝑡 ) = 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 i f i f
  • D 𝑓 ( 𝑡 ) = 1 1 ≤ 𝑡 ≤ 3 , 𝑡 − 3 3 < 𝑡 ≤ 5 i f i f , 𝑔 ( 𝑡 ) = 𝑡 1 ≤ 𝑡 ≤ 3 , 2 3 < 𝑡 ≤ 5 i f i f
  • E 𝑓 ( 𝑡 ) = 1 1 ≤ 𝑡 ≤ 3 , 𝑡 + 2 3 < 𝑡 ≤ 5 i f i f , 𝑔 ( 𝑡 ) = 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 i f i f

Q8:

Let 𝐴 = ( 1 , 1 ) and 𝐵 = ( 1 , 3 ) . Which of the following is a parameterization of 𝐴 𝐵 over 0 ≤ 𝑡 ≤ 1 that starts at 𝐴 and ends at 𝐵 .

  • A 𝑥 = 1 , 𝑦 = 2 ( 𝑡 + 1 )
  • B 𝑥 = 1 , 𝑦 = 𝑡 + 1
  • C 𝑥 = 𝑡 + 1 , 𝑦 = 1
  • D 𝑥 = 1 , 𝑦 = 2 𝑡 + 1
  • E 𝑥 = 2 𝑡 + 1 , 𝑦 = 1

Q9:

Let 𝐴 = ( 1 , 1 ) and 𝐵 = ( 1 , 2 ) . Which of the following is a parameterisation of 𝐴 𝐵 over 0 ≤ 𝑡 ≤ 1 that starts at 𝐵 and ends at 𝐴 .

  • A 𝑥 = 1 , 𝑦 = 2 + 𝑡
  • B 𝑥 = 1 , 𝑦 = 𝑡 + 1
  • C 𝑥 = 2 − 𝑡 , 𝑦 = 1
  • D 𝑥 = 1 , 𝑦 = 2 − 𝑡
  • E 𝑥 = 𝑡 + 1 , 𝑦 = 1

Q10:

Find the parameterization 𝑥 = 𝑓 ( 𝑡 ) , 𝑦 = 𝑔 ( 𝑡 ) of the path 𝐴 , 𝐵 , 𝐶 , 𝐷 using the interval 1 ≤ 𝑡 ≤ 9 .

  • A 𝑓 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 1 ≤ 𝑡 ≤ 3 , 𝑡 + 2 3 < 𝑡 ≤ 5 , 3 5 < 𝑡 ≤ 7 , 1 0 + 𝑡 7 < 𝑡 ≤ 9 i f i f i f i f , 𝑔 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 , 8 − 𝑡 5 < 𝑡 ≤ 7 , 1 7 < 𝑡 ≤ 9 i f i f i f i f
  • B 𝑓 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 1 ≤ 𝑡 ≤ 3 , 𝑡 − 2 3 < 𝑡 ≤ 5 , 3 5 < 𝑡 ≤ 7 , 𝑡 − 1 0 7 < 𝑡 ≤ 9 i f i f i f i f , 𝑔 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 , 𝑡 − 8 5 < 𝑡 ≤ 7 , 1 7 < 𝑡 ≤ 9 i f i f i f i f
  • C 𝑓 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 1 ≤ 𝑡 ≤ 3 , 𝑡 + 2 3 < 𝑡 ≤ 5 , 3 5 < 𝑡 ≤ 7 , 𝑡 + 1 0 7 < 𝑡 ≤ 9 i f i f i f i f , 𝑔 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 , 𝑡 − 8 5 < 𝑡 ≤ 7 , 1 7 < 𝑡 ≤ 9 i f i f i f i f
  • D 𝑓 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 1 ≤ 𝑡 ≤ 3 , 𝑡 − 2 3 < 𝑡 ≤ 5 , 3 5 < 𝑡 ≤ 7 , 1 0 − 𝑡 7 < 𝑡 ≤ 9 i f i f i f i f , 𝑔 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 , 8 − 𝑡 5 < 𝑡 ≤ 7 , 1 7 < 𝑡 ≤ 9 i f i f i f i f
  • E 𝑓 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 3 1 ≤ 𝑡 ≤ 3 , 𝑡 − 2 3 < 𝑡 ≤ 5 , 1 5 < 𝑡 ≤ 7 , 1 0 − 𝑡 7 < 𝑡 ≤ 9 i f i f i f i f , 𝑔 ( 𝑡 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑡 1 ≤ 𝑡 ≤ 3 , 3 3 < 𝑡 ≤ 5 , 8 − 𝑡 5 < 𝑡 ≤ 7 , 1 7 < 𝑡 ≤ 9 i f i f i f i f

Q11:

Let 𝐴 = ( 1 , 1 ) and 𝐵 = ( 1 , 2 ) . Find the parameterisation of 𝐴 𝐵 over 0 ≤ 𝑡 ≤ 1 that starts at 𝐴 and ends at 𝐵 .

  • A 𝑥 = 1 , 𝑦 = 𝑡 − 1
  • B 𝑥 = 1 , 𝑦 = 𝑡
  • C 𝑥 = 𝑡 + 1 , 𝑦 = 𝑡
  • D 𝑥 = 1 , 𝑦 = 𝑡 + 1
  • E 𝑥 = 𝑡 + 1 , 𝑦 = 𝑡

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 𝑥 = 3 + 𝑡 , 𝑦 = 3 + 𝑡 , 𝑧 = 3 + 𝑡
  • B 𝑥 = 3 , 𝑦 = 3 , 𝑧 = 3
  • C 𝑥 = 1 + 3 𝑡 , 𝑦 = 1 + 3 𝑡 , 𝑧 = 1 + 3 𝑡
  • D 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 , 𝑧 = 3 𝑡

Q13:

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

  • Afalse
  • Btrue

Q14:

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

  • A 𝑥 = 9 − 7 𝐾 , 𝑦 = 3 + 2 𝐾
  • B 𝑥 = − 7 + 2 𝐾 , 𝑦 = 9 + 3 𝐾
  • C 𝑥 = − 7 + 3 𝐾 , 𝑦 = 9 + 2 𝐾
  • D 𝑥 = 9 + 3 𝐾 , 𝑦 = − 7 + 2 𝐾

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