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 (โˆ’9,8) with direction vector (4,โˆ’7).

  • A๐‘ฅ=8โˆ’7๐พ, ๐‘ฆ=โˆ’9+4๐พ
  • B๐‘ฅ=โˆ’9+8๐พ, ๐‘ฆ=4โˆ’7๐พ
  • C๐‘ฅ=8+4๐พ, ๐‘ฆ=โˆ’9โˆ’7๐พ
  • D๐‘ฅ=โˆ’9+4๐พ, ๐‘ฆ=8โˆ’7๐พ

Q2:

Find the parametric equations of the straight line that makes an angle of 135โˆ˜ with the positive ๐‘ฅ-axis and passes through the point (1,โˆ’15).

  • A๐‘ฅ=1+๐พ, ๐‘ฆ=1โˆ’15๐พ
  • B๐‘ฅ=โˆ’15โˆ’๐พ, ๐‘ฆ=1+๐พ
  • C๐‘ฅ=1, ๐‘ฆ=โˆ’15โˆ’๐พ
  • D๐‘ฅ=1+๐พ, ๐‘ฆ=โˆ’15โˆ’๐พ

Q3:

Consider the line shown that passes through the point (3,4) and makes an angle of 45 degrees with the positive ๐‘ฅ-axis.

Suppose that the distance between (3,4) and any point (๐‘ฅ,๐‘ฆ) on the line is ๐‘Ÿ.

Write, in terms of ๐‘Ÿ, an expression for the horizontal distance ๐‘ฅโˆ’3 between the two points.

  • A๐‘Ÿ2
  • B๐‘Ÿโˆš2
  • C๐‘Ÿ
  • D๐‘Ÿ2๏Šจ

Write, in terms of ๐‘Ÿ, an expression for the vertical distance ๐‘ฆโˆ’4 between the two points.

  • A๐‘Ÿ
  • B๐‘Ÿ2
  • C๐‘Ÿ2๏Šจ
  • D๐‘Ÿโˆš2

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

  • A๐‘ฅ=3+๐‘Ÿโˆš2, ๐‘ฆ=4โˆ’๐‘Ÿโˆš2
  • B๐‘ฅ=3โˆ’๐‘Ÿโˆš2, ๐‘ฆ=4โˆ’๐‘Ÿโˆš2
  • C๐‘ฅ=3+๐‘Ÿโˆš2, ๐‘ฆ=4+๐‘Ÿโˆš2
  • D๐‘ฅ=3โˆ’๐‘Ÿโˆš2, ๐‘ฆ=4+๐‘Ÿโˆš2

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

  • A(3โˆ’2โˆš2,4โˆ’2โˆš2)
  • B(3โˆ’2โˆš2,4+2โˆš2)
  • C(3+2โˆš2,4โˆ’2โˆš2)
  • D(3+2โˆš2,4+2โˆš2)

Q4:

Write a pair of parametric equations with parameter ๐‘Ÿ describing the shown line.

  • A๐‘ฅ=2+๐‘Ÿ2, ๐‘ฆ=3โˆ’๐‘Ÿโˆš32
  • B๐‘ฅ=2โˆ’๐‘Ÿ2, ๐‘ฆ=3+๐‘Ÿโˆš32
  • C๐‘ฅ=2โˆ’๐‘Ÿ2, ๐‘ฆ=3โˆ’๐‘Ÿโˆš32
  • D๐‘ฅ=2+๐‘Ÿโˆš32,๐‘ฆ=3+๐‘Ÿ2

Q5:

Write the parametric equation of the straight line that passes through the point (๐‘Ž,๐‘) and makes an angle of ๐œƒ with the positive ๐‘ฅ-axis as shown.

  • A๐‘Ÿ=๐‘ฅโˆ’๐‘Ž(๐œƒ)=๐‘ฆโˆ’๐‘(๐œƒ)coscos
  • B๐‘Ÿ=๐‘ฅ+๐‘Ž(๐œƒ)=๐‘ฆ+๐‘(๐œƒ)cossin
  • C๐‘Ÿ=๐‘ฅโˆ’๐‘Ž(๐œƒ)=๐‘ฆโˆ’๐‘(๐œƒ)cossin
  • D๐‘Ÿ=๐‘ฅโˆ’๐‘Ž(๐œƒ)=๐‘ฆโˆ’๐‘(๐œƒ)sinsin

Q6:

The equations ๐‘ฅ=2๐‘ก+1,๐‘ฆ=โˆ’3๐‘ก+2 parameterize the line segment between (1,2) and (3,โˆ’1) over the interval 0โ‰ค๐‘กโ‰ค1.

Which of the following is a parameterization of the line segment on โˆ’1โ‰ค๐‘กโ‰ค0?

  • A๐‘ฅ=2๐‘ก+3,๐‘ฆ=3๐‘ก+1
  • B๐‘ฅ=3๐‘กโˆ’2,๐‘ฆ=โˆ’๐‘กโˆ’3
  • C๐‘ฅ=โˆ’2๐‘ก+3,๐‘ฆ=3๐‘กโˆ’1
  • D๐‘ฅ=2๐‘ก+3,๐‘ฆ=โˆ’3๐‘กโˆ’1
  • E๐‘ฅ=โˆ’2๐‘กโˆ’3,๐‘ฆ=โˆ’3๐‘ก+1

Which of the following is a parameterization of the line segment on 0โ‰ค๐‘กโ‰ค1 that starts at (3,โˆ’1) and ends at (1,2)?

  • A๐‘ฅ=3๐‘กโˆ’2,๐‘ฆ=โˆ’๐‘กโˆ’3
  • B๐‘ฅ=2๐‘ก+3,๐‘ฆ=3๐‘ก+1
  • C๐‘ฅ=โˆ’2๐‘กโˆ’3,๐‘ฆ=โˆ’3๐‘ก+1
  • D๐‘ฅ=2๐‘ก+3,๐‘ฆ=โˆ’3๐‘กโˆ’1
  • E๐‘ฅ=โˆ’2๐‘ก+3,๐‘ฆ=3๐‘กโˆ’1

Which of the following is a parameterization of the line segment on 0โ‰ค๐‘กโ‰ค2?

  • A๐‘ฅ=๐‘กโˆ’1,๐‘ฆ=โˆ’32๐‘กโˆ’2
  • B๐‘ฅ=๐‘กโˆ’1,๐‘ฆ=โˆ’32๐‘ก+2
  • C๐‘ฅ=๐‘ก+1,๐‘ฆ=โˆ’32๐‘ก+2
  • D๐‘ฅ=4๐‘ก+1,๐‘ฆ=โˆ’6๐‘ก+2
  • E๐‘ฅ=4๐‘กโˆ’1,๐‘ฆ=โˆ’6๐‘กโˆ’2

If the parameterizations you have given above correspond to a particle moving along the line segment, how does the parameterization over interval 0โ‰ค๐‘กโ‰ค2 relate to the one over 0โ‰ค๐‘กโ‰ค1?

  • AOver 0โ‰ค๐‘กโ‰ค2, the particle is moving twice as fast as over 0โ‰ค๐‘กโ‰ค1.
  • BOver 0โ‰ค๐‘กโ‰ค2, the particle is moving one-third as fast as over 0โ‰ค๐‘กโ‰ค1.
  • COver 0โ‰ค๐‘กโ‰ค2, the particle is moving half as fast as over 0โ‰ค๐‘กโ‰ค1.
  • DOver 0โ‰ค๐‘กโ‰ค2, the particle is moving three times as fast as over 0โ‰ค๐‘กโ‰ค1.

Q7:

Consider the points ๐ด,๐ต, and ๐ถ and the line segments in the figure.

Give the parameterization of ๐ด๐ต over the interval 1โ‰ค๐‘กโ‰ค3.

  • A๐‘ฅ=1, ๐‘ฆ=2๐‘กโˆ’1
  • B๐‘ฅ=๐‘ก, ๐‘ฆ=0
  • C๐‘ฅ=๐‘ก, ๐‘ฆ=1
  • D๐‘ฅ=0, ๐‘ฆ=๐‘ก
  • E๐‘ฅ=1, ๐‘ฆ=๐‘ก

Give the parameterization of ๐ต๐ถ over the interval 3โ‰ค๐‘กโ‰ค5.

  • A๐‘ฅ=2, ๐‘ฆ=3
  • B๐‘ฅ=๐‘กโˆ’3, ๐‘ฆ=2
  • C๐‘ฅ=3, ๐‘ฆ=๐‘กโˆ’2
  • D๐‘ฅ=๐‘กโˆ’2, ๐‘ฆ=3
  • E๐‘ฅ=2, ๐‘ฆ=๐‘กโˆ’3

Find functions ๐‘“ and ๐‘” defined for 1โ‰ค๐‘กโ‰ค5 so that ๐‘ฅ=๐‘“(๐‘ก), ๐‘ฆ=๐‘”(๐‘ก) parameterizes the given path from ๐ด to ๐ถ.

  • A๐‘“(๐‘ก)=๏ญ11โ‰ค๐‘กโ‰ค3,๐‘กโˆ’23<๐‘กโ‰ค5ifif, ๐‘”(๐‘ก)=๏ญ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5ifif
  • B๐‘“(๐‘ก)=๏ญ11โ‰ค๐‘กโ‰ค3,๐‘กโˆ’33<๐‘กโ‰ค5ifif, ๐‘”(๐‘ก)=๏ญ๐‘ก1โ‰ค๐‘กโ‰ค3,23<๐‘กโ‰ค5ifif
  • C๐‘“(๐‘ก)=๏ญ๐‘ก1โ‰ค๐‘กโ‰ค3,23<๐‘กโ‰ค5ifif, ๐‘”(๐‘ก)=๏ญ11โ‰ค๐‘กโ‰ค3,๐‘กโˆ’33<๐‘กโ‰ค5ifif
  • D๐‘“(๐‘ก)=๏ญ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5ifif, ๐‘”(๐‘ก)=๏ญ11โ‰ค๐‘กโ‰ค3,๐‘กโˆ’23<๐‘กโ‰ค5ifif
  • E๐‘“(๐‘ก)=๏ญ11โ‰ค๐‘กโ‰ค3,๐‘ก+23<๐‘กโ‰ค5ifif, ๐‘”(๐‘ก)=๏ญ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5ifif

Q8:

Let ๐ด=(1,1) and ๐ต=(1,3). Which of the following is a parameterization of ๐ด๐ต over 0โ‰ค๐‘กโ‰ค1 that starts at ๐ด and ends at ๐ต?

  • A๐‘ฅ=1, ๐‘ฆ=๐‘ก+1
  • B๐‘ฅ=๐‘ก+1, ๐‘ฆ=1
  • C๐‘ฅ=1, ๐‘ฆ=2(๐‘ก+1)
  • D๐‘ฅ=1, ๐‘ฆ=2๐‘ก+1
  • E๐‘ฅ=2๐‘ก+1, ๐‘ฆ=1

Q9:

Let ๐ด=(1,1) and ๐ต=(1,2). Which of the following is a parameterization of ๐ด๐ต over 0โ‰ค๐‘กโ‰ค1 that starts at ๐ต and ends at ๐ด?

  • A๐‘ฅ=2โˆ’๐‘ก, ๐‘ฆ=1
  • B๐‘ฅ=1, ๐‘ฆ=2โˆ’๐‘ก
  • C๐‘ฅ=1, ๐‘ฆ=2+๐‘ก
  • D๐‘ฅ=1, ๐‘ฆ=๐‘ก+1
  • E๐‘ฅ=๐‘ก+1, ๐‘ฆ=1

Q10:

Find the parameterization ๐‘ฅ=๐‘“(๐‘ก), ๐‘ฆ=๐‘”(๐‘ก) of the path ๐ด,๐ต,๐ถ,๐ท using the interval 1โ‰ค๐‘กโ‰ค9.

  • A๐‘“(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ11โ‰ค๐‘กโ‰ค3,๐‘กโˆ’23<๐‘กโ‰ค5,35<๐‘กโ‰ค7,10โˆ’๐‘ก7<๐‘กโ‰ค9ifififif, ๐‘”(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5,8โˆ’๐‘ก5<๐‘กโ‰ค7,17<๐‘กโ‰ค9ifififif
  • B๐‘“(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ11โ‰ค๐‘กโ‰ค3,๐‘ก+23<๐‘กโ‰ค5,35<๐‘กโ‰ค7,10+๐‘ก7<๐‘กโ‰ค9ifififif, ๐‘”(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5,8โˆ’๐‘ก5<๐‘กโ‰ค7,17<๐‘กโ‰ค9ifififif
  • C๐‘“(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ11โ‰ค๐‘กโ‰ค3,๐‘กโˆ’23<๐‘กโ‰ค5,35<๐‘กโ‰ค7,๐‘กโˆ’107<๐‘กโ‰ค9ifififif, ๐‘”(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5,๐‘กโˆ’85<๐‘กโ‰ค7,17<๐‘กโ‰ค9ifififif
  • D๐‘“(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ31โ‰ค๐‘กโ‰ค3,๐‘กโˆ’23<๐‘กโ‰ค5,15<๐‘กโ‰ค7,10โˆ’๐‘ก7<๐‘กโ‰ค9ifififif, ๐‘”(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5,8โˆ’๐‘ก5<๐‘กโ‰ค7,17<๐‘กโ‰ค9ifififif
  • E๐‘“(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ11โ‰ค๐‘กโ‰ค3,๐‘ก+23<๐‘กโ‰ค5,35<๐‘กโ‰ค7,๐‘ก+107<๐‘กโ‰ค9ifififif, ๐‘”(๐‘ก)=โŽงโŽชโŽจโŽชโŽฉ๐‘ก1โ‰ค๐‘กโ‰ค3,33<๐‘กโ‰ค5,๐‘กโˆ’85<๐‘กโ‰ค7,17<๐‘กโ‰ค9ifififif

Q11:

Let ๐ด=(1,1) and ๐ต=(1,2). Find the parameterization of ๐ด๐ต over 0โ‰ค๐‘กโ‰ค1 that starts at ๐ด and ends at ๐ต.

  • A๐‘ฅ=๐‘ก+1,๐‘ฆ=๐‘ก
  • B๐‘ฅ=๐‘ก+1,๐‘ฆ=๐‘ก
  • C๐‘ฅ=1,๐‘ฆ=๐‘ก
  • D๐‘ฅ=1,๐‘ฆ=๐‘ก+1
  • E๐‘ฅ=1,๐‘ฆ=๐‘กโˆ’1

Q12:

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

  • AFalse
  • BTrue

Q13:

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

  • A๐‘ฅ=โˆ’7+2๐พ, ๐‘ฆ=9+3๐พ
  • B๐‘ฅ=9โˆ’7๐พ, ๐‘ฆ=3+2๐พ
  • C๐‘ฅ=โˆ’7+3๐พ, ๐‘ฆ=9+2๐พ
  • D๐‘ฅ=9+3๐พ, ๐‘ฆ=โˆ’7+2๐พ

Q14:

The lines ๐‘ฅ=3๐‘กโˆ’2๏Šง, ๐‘ฆ=3๐‘ก+2๏Šง, ๐‘ง=9๐‘กโˆ’2๏Šง and ๐‘ฅ=๐‘Ž๐‘กโˆ’2๏Šจ, ๐‘ฆ=๐‘ก+1๏Šจ, ๐‘ง=๐‘๐‘กโˆ’2๏Šจ are parallel. What is ๐‘Ž+๐‘?

Q15:

In the following figure, if the equation of the straight line โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐ด๐ต is ๐‘ฅ10+๐‘ฆ12=1, then the parametric equation of the straight line โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ท is .

  • A๐‘ฅ=6+6๐‘˜, ๐‘ฆ=5+5๐‘˜
  • B๐‘ฅ=6+5๐‘˜, ๐‘ฆ=5+6๐‘˜
  • C๐‘ฅ=5+5๐‘˜, ๐‘ฆ=6+6๐‘˜
  • D๐‘ฅ=5+6๐‘˜, ๐‘ฆ=6+5๐‘˜

Q16:

Straight line ๐ฟ passes through the point ๐‘(3,4) and has a direction vector u=โŸจ2,โˆ’5โŸฉ. Then, the parametric equations of line ๐ฟ are .

  • A๐‘ฅ=โˆ’2โˆ’3๐‘˜, ๐‘ฆ=5+4๐‘˜
  • B๐‘ฅ=โˆ’3โˆ’2๐‘˜, ๐‘ฆ=โˆ’4+5๐‘˜
  • C๐‘ฅ=2+3๐‘˜, ๐‘ฆ=โˆ’5+4๐‘˜
  • D๐‘ฅ=3+2๐‘˜, ๐‘ฆ=4โˆ’5๐‘˜

Q17:

The direction vector of the straight line whose parametric equations are ๐‘‹=2 and ๐‘Œ=โˆ’2๐‘˜+4 is .

  • AโŸจ0,4โŸฉ
  • BโŸจ0,โˆ’2โŸฉ
  • CโŸจ2,4โŸฉ
  • DโŸจ2,โˆ’2โŸฉ

Q18:

Which of the following are the parametric equations of the line through point ๐ด(โˆ’8,8) with a direction perpendicular to vector u=โŸจโˆ’6,7โŸฉ?

  • A๐‘ฅ=โˆ’8+7๐พ, ๐‘ฆ=8โˆ’6๐พ
  • B๐‘ฅ=โˆ’8+8๐พ, ๐‘ฆ=โˆ’6+7๐พ
  • C๐‘ฅ=โˆ’8+7๐พ, ๐‘ฆ=8+6๐พ
  • D๐‘ฅ=โˆ’8โˆ’6๐พ, ๐‘ฆ=8+7๐พ

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