Worksheet: Equation of a Straight Line: Parametric Form

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).

  • A𝑥=8−7𝐾, 𝑦=−9+4𝐾
  • B𝑥=−9+8𝐾, 𝑦=4−7𝐾
  • C𝑥=8+4𝐾, 𝑦=−9−7𝐾
  • D𝑥=−9+4𝐾, 𝑦=8−7𝐾

Q2:

Find the parametric equations of the straight line that makes an angle of 135∘ with the positive 𝑥-axis and passes through the point (1,−15).

  • A𝑥=1+𝐾, 𝑦=1−15𝐾
  • B𝑥=−15−𝐾, 𝑦=1+𝐾
  • C𝑥=1, 𝑦=−15−𝐾
  • D𝑥=1+𝐾, 𝑦=−15−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.

  • A𝑟2
  • B𝑟√2
  • C𝑟
  • D𝑟2

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

  • A𝑟
  • B𝑟2
  • C𝑟2
  • D𝑟√2

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

  • A𝑥=3+𝑟√2, 𝑦=4−𝑟√2
  • B𝑥=3−𝑟√2, 𝑦=4−𝑟√2
  • C𝑥=3+𝑟√2, 𝑦=4+𝑟√2
  • D𝑥=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.

  • A𝑥=2+𝑟2, 𝑦=3−𝑟√32
  • B𝑥=2−𝑟2, 𝑦=3+𝑟√32
  • C𝑥=2−𝑟2, 𝑦=3−𝑟√32
  • D𝑥=2+𝑟√32,𝑦=3+𝑟2

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.

  • A𝑟=𝑥−𝑎(𝜃)=𝑦−𝑏(𝜃)coscos
  • B𝑟=𝑥+𝑎(𝜃)=𝑦+𝑏(𝜃)cossin
  • C𝑟=𝑥−𝑎(𝜃)=𝑦−𝑏(𝜃)cossin
  • D𝑟=𝑥−𝑎(𝜃)=𝑦−𝑏(𝜃)sinsin

Q6:

The equations 𝑥=2𝑡+1,𝑦=−3𝑡+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?

  • A𝑥=2𝑡+3,𝑦=3𝑡+1
  • B𝑥=3𝑡−2,𝑦=−𝑡−3
  • C𝑥=−2𝑡+3,𝑦=3𝑡−1
  • D𝑥=2𝑡+3,𝑦=−3𝑡−1
  • E𝑥=−2𝑡−3,𝑦=−3𝑡+1

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)?

  • A𝑥=3𝑡−2,𝑦=−𝑡−3
  • B𝑥=2𝑡+3,𝑦=3𝑡+1
  • C𝑥=−2𝑡−3,𝑦=−3𝑡+1
  • D𝑥=2𝑡+3,𝑦=−3𝑡−1
  • E𝑥=−2𝑡+3,𝑦=3𝑡−1

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

  • A𝑥=𝑡−1,𝑦=−32𝑡−2
  • B𝑥=𝑡−1,𝑦=−32𝑡+2
  • C𝑥=𝑡+1,𝑦=−32𝑡+2
  • D𝑥=4𝑡+1,𝑦=−6𝑡+2
  • E𝑥=4𝑡−1,𝑦=−6𝑡−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 twice as fast as over 0≤𝑡≤1.
  • BOver 0≤𝑡≤2, the particle is moving one-third as fast as over 0≤𝑡≤1.
  • COver 0≤𝑡≤2, the particle is moving half as fast as over 0≤𝑡≤1.
  • DOver 0≤𝑡≤2, the particle is moving three times 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 𝐴𝐵 over the interval 1≤𝑡≤3.

  • A𝑥=1, 𝑦=2𝑡−1
  • B𝑥=𝑡, 𝑦=0
  • C𝑥=𝑡, 𝑦=1
  • D𝑥=0, 𝑦=𝑡
  • E𝑥=1, 𝑦=𝑡

Give the parameterization of 𝐵𝐶 over the interval 3≤𝑡≤5.

  • A𝑥=2, 𝑦=3
  • B𝑥=𝑡−3, 𝑦=2
  • C𝑥=3, 𝑦=𝑡−2
  • D𝑥=𝑡−2, 𝑦=3
  • E𝑥=2, 𝑦=𝑡−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.

  • A𝑓(𝑡)=11≤𝑡≤3,𝑡−23<𝑡≤5ifif, 𝑔(𝑡)=𝑡1≤𝑡≤3,33<𝑡≤5ifif
  • B𝑓(𝑡)=11≤𝑡≤3,𝑡−33<𝑡≤5ifif, 𝑔(𝑡)=𝑡1≤𝑡≤3,23<𝑡≤5ifif
  • C𝑓(𝑡)=𝑡1≤𝑡≤3,23<𝑡≤5ifif, 𝑔(𝑡)=11≤𝑡≤3,𝑡−33<𝑡≤5ifif
  • D𝑓(𝑡)=𝑡1≤𝑡≤3,33<𝑡≤5ifif, 𝑔(𝑡)=11≤𝑡≤3,𝑡−23<𝑡≤5ifif
  • E𝑓(𝑡)=11≤𝑡≤3,𝑡+23<𝑡≤5ifif, 𝑔(𝑡)=𝑡1≤𝑡≤3,33<𝑡≤5ifif

Q8:

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

  • A𝑥=1, 𝑦=𝑡+1
  • B𝑥=𝑡+1, 𝑦=1
  • C𝑥=1, 𝑦=2(𝑡+1)
  • D𝑥=1, 𝑦=2𝑡+1
  • E𝑥=2𝑡+1, 𝑦=1

Q9:

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

  • A𝑥=2−𝑡, 𝑦=1
  • B𝑥=1, 𝑦=2−𝑡
  • C𝑥=1, 𝑦=2+𝑡
  • D𝑥=1, 𝑦=𝑡+1
  • E𝑥=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.

  • A𝑓(𝑡)=⎧⎪⎨⎪⎩11≤𝑡≤3,𝑡−23<𝑡≤5,35<𝑡≤7,10−𝑡7<𝑡≤9ifififif, 𝑔(𝑡)=⎧⎪⎨⎪⎩𝑡1≤𝑡≤3,33<𝑡≤5,8−𝑡5<𝑡≤7,17<𝑡≤9ifififif
  • B𝑓(𝑡)=⎧⎪⎨⎪⎩11≤𝑡≤3,𝑡+23<𝑡≤5,35<𝑡≤7,10+𝑡7<𝑡≤9ifififif, 𝑔(𝑡)=⎧⎪⎨⎪⎩𝑡1≤𝑡≤3,33<𝑡≤5,8−𝑡5<𝑡≤7,17<𝑡≤9ifififif
  • C𝑓(𝑡)=⎧⎪⎨⎪⎩11≤𝑡≤3,𝑡−23<𝑡≤5,35<𝑡≤7,𝑡−107<𝑡≤9ifififif, 𝑔(𝑡)=⎧⎪⎨⎪⎩𝑡1≤𝑡≤3,33<𝑡≤5,𝑡−85<𝑡≤7,17<𝑡≤9ifififif
  • D𝑓(𝑡)=⎧⎪⎨⎪⎩31≤𝑡≤3,𝑡−23<𝑡≤5,15<𝑡≤7,10−𝑡7<𝑡≤9ifififif, 𝑔(𝑡)=⎧⎪⎨⎪⎩𝑡1≤𝑡≤3,33<𝑡≤5,8−𝑡5<𝑡≤7,17<𝑡≤9ifififif
  • E𝑓(𝑡)=⎧⎪⎨⎪⎩11≤𝑡≤3,𝑡+23<𝑡≤5,35<𝑡≤7,𝑡+107<𝑡≤9ifififif, 𝑔(𝑡)=⎧⎪⎨⎪⎩𝑡1≤𝑡≤3,33<𝑡≤5,𝑡−85<𝑡≤7,17<𝑡≤9ifififif

Q11:

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

  • A𝑥=𝑡+1,𝑦=𝑡
  • B𝑥=𝑡+1,𝑦=𝑡
  • C𝑥=1,𝑦=𝑡
  • D𝑥=1,𝑦=𝑡+1
  • E𝑥=1,𝑦=𝑡−1

Q12:

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

  • AFalse
  • BTrue

Q13:

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

  • A𝑥=−7+2𝐾, 𝑦=9+3𝐾
  • B𝑥=9−7𝐾, 𝑦=3+2𝐾
  • C𝑥=−7+3𝐾, 𝑦=9+2𝐾
  • D𝑥=9+3𝐾, 𝑦=−7+2𝐾

Q14:

The lines 𝑥=3𝑡−2, 𝑦=3𝑡+2, 𝑧=9𝑡−2 and 𝑥=𝑎𝑡−2, 𝑦=𝑡+1, 𝑧=𝑏𝑡−2 are parallel. What is 𝑎+𝑏?

Q15:

In the following figure, if the equation of the straight line ⃖⃗𝐴𝐵 is 𝑥10+𝑦12=1, then the parametric equation of the straight line ⃖⃗𝑂𝐷 is .

  • A𝑥=6+6𝑘, 𝑦=5+5𝑘
  • B𝑥=6+5𝑘, 𝑦=5+6𝑘
  • C𝑥=5+5𝑘, 𝑦=6+6𝑘
  • D𝑥=5+6𝑘, 𝑦=6+5𝑘

Q16:

Straight line 𝐿 passes through the point 𝑁(3,4) and has a direction vector u=⟨2,−5⟩. Then, the parametric equations of line 𝐿 are .

  • A𝑥=−2−3𝑘, 𝑦=5+4𝑘
  • B𝑥=−3−2𝑘, 𝑦=−4+5𝑘
  • C𝑥=2+3𝑘, 𝑦=−5+4𝑘
  • D𝑥=3+2𝑘, 𝑦=4−5𝑘

Q17:

The direction vector of the straight line whose parametric equations are 𝑋=2 and 𝑌=−2𝑘+4 is .

  • A⟨0,4⟩
  • B⟨0,−2⟩
  • C⟨2,4⟩
  • D⟨2,−2⟩

Q18:

Which of the following are the parametric equations of the line through point 𝐴(−8,8) with a direction perpendicular to vector u=⟨−6,7⟩?

  • A𝑥=−8+7𝐾, 𝑦=8−6𝐾
  • B𝑥=−8+8𝐾, 𝑦=−6+7𝐾
  • C𝑥=−8+7𝐾, 𝑦=8+6𝐾
  • D𝑥=−8−6𝐾, 𝑦=8+7𝐾

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