Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.
Start Practicing

Worksheet: Finding the Tangent and the Normal Equations of a Parametrically Defined Curve

Q1:

Find the slope of the tangent to the astroid π‘₯ = π‘Ž πœƒ c o s 3 , 𝑦 = π‘Ž πœƒ s i n 3 in terms of πœƒ .

  • A βˆ’ πœƒ t a n 2
  • B βˆ’ πœƒ c o t
  • C c o t 2 πœƒ
  • D βˆ’ πœƒ t a n
  • E t a n πœƒ

Q2:

Find an equation of the tangent to the curve π‘₯ = 1 + 𝑑 l n , 𝑦 = 𝑑 + 2 2 at the point ( 1 , 3 ) .

  • A 𝑦 = π‘₯ + 2
  • B 𝑦 = 1 2 π‘₯ + 5 2
  • C 𝑦 = 2 π‘₯ βˆ’ 1
  • D 𝑦 = 2 π‘₯ + 1
  • E 𝑦 = 1 2 π‘₯ βˆ’ 5 2

Q3:

Find an equation of the tangent to the curve π‘₯ = 𝑑 βˆ’ 𝑑 2 , 𝑦 = 𝑑 + 𝑑 + 1 2 at the point ( 0 , 3 ) .

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

Q4:

Find an equation of the tangent to the curve π‘₯ = 1 + √ 𝑑 , 𝑦 = 𝑒 𝑑 2 at the point ( 2 , 𝑒 ) .

  • A 𝑦 = 𝑒 π‘₯ βˆ’ 𝑒
  • B 𝑦 = 2 𝑒 π‘₯ βˆ’ 3 𝑒
  • C 𝑦 = 4 𝑒 π‘₯ + 9 𝑒
  • D 𝑦 = 4 𝑒 π‘₯ βˆ’ 7 𝑒
  • E 𝑦 = 2 𝑒 π‘₯ + 4 𝑒

Q5:

Find an equation of the tangent to the curve π‘₯ = √ 𝑑 , 𝑦 = 𝑑 βˆ’ 2 𝑑 2 at the point corresponding to the value 𝑑 = 4 .

  • A 𝑦 = 3 2 π‘₯ βˆ’ 5 6
  • B 𝑦 = 1 2 π‘₯ βˆ’ 1 6
  • C 𝑦 = 1 6 π‘₯ βˆ’ 2 4
  • D 𝑦 = 2 4 π‘₯ βˆ’ 4 0
  • E 𝑦 = 3 0 π‘₯ βˆ’ 5 2

Q6:

Find the equation of the tangent to the curve π‘₯ = 𝑑 + 1 3 , 𝑦 = 𝑑 + 𝑑 4 at the point corresponding to the value 𝑑 = βˆ’ 1 .

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

Q7:

Find an equation of the tangent to the curve π‘₯ = 𝑑 𝑑 c o s , 𝑦 = 𝑑 𝑑 s i n at the point corresponding to the value 𝑑 = πœ‹ .

  • A 𝑦 = πœ‹ π‘₯ + πœ‹
  • B 𝑦 = βˆ’ πœ‹ π‘₯ + πœ‹ 2
  • C 𝑦 = βˆ’ πœ‹ π‘₯ βˆ’ πœ‹ 2
  • D 𝑦 = πœ‹ π‘₯ + πœ‹ 2
  • E 𝑦 = πœ‹ π‘₯ βˆ’ πœ‹

Q8:

Find an equation of the tangent to the curve π‘₯ = 𝑒 πœ‹ 𝑑 𝑑 s i n , 𝑦 = 𝑒 2 𝑑 at the point corresponding to the value 𝑑 = 0 .

  • A 𝑦 = π‘₯ + 1
  • B 𝑦 = 1 πœ‹ π‘₯ + 1
  • C 𝑦 = πœ‹ 2 π‘₯ + 1
  • D 𝑦 = 2 πœ‹ π‘₯ + 1
  • E 𝑦 = 2 π‘₯ + 1

Q9:

Given that π‘₯ = 2 𝑑 βˆ’ 9 3 , and 𝑦 = √ 7 𝑑 + 8 3 , determine the equation of the tangent to the curve at 𝑑 = βˆ’ 1 .

  • A βˆ’ 7 π‘₯ + 1 8 𝑦 + 5 9 = 0
  • B 7 π‘₯ + 1 8 𝑦 + 5 9 = 0
  • C βˆ’ 7 π‘₯ + 1 8 𝑦 βˆ’ 1 2 = 0
  • D βˆ’ 7 π‘₯ + 1 8 𝑦 βˆ’ 9 5 = 0

Q10:

Determine the equation of the normal to the curve π‘₯ = βˆ’ 4 πœƒ + 3 c o t , 𝑦 = 3 πœƒ + √ 2 πœƒ s i n s e c 2 at πœƒ = πœ‹ 4 .

  • A βˆ’ π‘₯ + 5 𝑦 8 βˆ’ 5 1 1 6 = 0
  • B π‘₯ + 8 𝑦 5 βˆ’ 2 3 5 = 0
  • C βˆ’ 5 π‘₯ 8 + 𝑦 βˆ’ 3 3 8 = 0
  • D π‘₯ + 5 𝑦 8 βˆ’ 1 9 1 6 = 0

Q11:

Find the equation of the normal to the curves π‘₯ = 2 πœƒ s e c and 𝑦 = 4 πœƒ t a n at πœƒ = πœ‹ 6 .

  • A 4 𝑦 βˆ’ π‘₯ βˆ’ 4 √ 3 = 0
  • B 𝑦 + 4 π‘₯ βˆ’ 2 0 √ 3 3 = 0
  • C 𝑦 βˆ’ 4 π‘₯ + 4 √ 3 = 0
  • D βˆ’ 4 𝑦 βˆ’ π‘₯ + 2 0 √ 3 3 = 0

Q12:

A curve 𝐢 is defined by the parametric equations π‘₯ = 𝑑 2 and 𝑦 = 𝑑 βˆ’ 3 𝑑 3 .

Find the two equations of the tangents to curve 𝐢 on the point at ( 3 , 0 ) .

  • A 𝑦 = √ 3 ο€Ή π‘₯ βˆ’ 3  2 , 𝑦 = βˆ’ √ 3 ο€Ή π‘₯ βˆ’ 3  2
  • B 𝑦 = 1 √ 3 ( π‘₯ βˆ’ 3 ) , 𝑦 = βˆ’ 1 √ 3 ( π‘₯ βˆ’ 3 )
  • C 𝑦 = 1 √ 3 ο€Ή π‘₯ βˆ’ 3  2 , 𝑦 = βˆ’ 1 √ 3 ο€Ή π‘₯ βˆ’ 3  2
  • D 𝑦 = √ 3 ( π‘₯ βˆ’ 3 ) , 𝑦 = βˆ’ √ 3 ( π‘₯ βˆ’ 3 )
  • E 𝑦 = 6 ο€Ή π‘₯ βˆ’ 3  2 , 𝑦 = βˆ’ 6 ο€Ή π‘₯ βˆ’ 3  2

Find all possible points on 𝐢 where the tangent is horizontal.

  • A ( 1 , 2 ) , ( 1 , βˆ’ 2 )
  • B ( 3 , 0 )
  • C ( 0 , 0 ) , ( 1 , 2 ) , ( 1 , βˆ’ 2 )
  • D ( 0 , 0 )
  • E ( 0 , 0 ) , ( 3 , 0 )

Find all possible points on 𝐢 where the tangent is vertical.

  • A ( 0 , 0 ) , ( 3 , 0 )
  • B ( 1 , 2 ) , ( 1 , βˆ’ 2 )
  • C ( 0 , 0 )
  • D ( 0 , 0 ) , ( 1 , 2 ) , ( 1 , βˆ’ 2 )
  • E ( 3 , 0 )

Q13:

Consider the curve π‘₯ = 𝑑 βˆ’ 3 𝑑 3 , 𝑦 = 𝑑 βˆ’ 3 𝑑 3 2 .

Find all points on this curve where the tangent is horizontal.

  • A ( 2 , βˆ’ 4 ) , ( 0 , 0 ) , ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , βˆ’ 2 )
  • B ( βˆ’ 2 , βˆ’ 2 ) , ( 2 , βˆ’ 4 )
  • C ( 0 , 0 )
  • D ( 2 , βˆ’ 4 ) , ( 0 , 0 )
  • EThere are no horizontal tangents.

Find all points on this curve where the tangent is vertical.

  • A ( βˆ’ 2 , βˆ’ 2 ) , ( 2 , βˆ’ 4 )
  • B ( βˆ’ 2 , βˆ’ 2 )
  • C ( 2 , βˆ’ 4 ) , ( 0 , 0 ) , ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , βˆ’ 2 )
  • D ( 2 , βˆ’ 4 ) , ( 0 , 0 )
  • EThere are no vertical tangents.

Q14:

Consider the curve π‘₯ = 𝑒 s i n πœƒ , 𝑦 = 𝑒 c o s πœƒ .

Find all points on this curve where the tangent is horizontal.

  • A ο€Ή 1 , 𝑒  βˆ’ 1
  • B ( 𝑒 , 1 ) , ο€Ή 𝑒 , 1  βˆ’ 1
  • C ( 1 , 𝑒 ) , ο€Ή 1 , 𝑒  βˆ’ 1 , ( 𝑒 , 1 ) , ο€Ή 𝑒 , 1  βˆ’ 1
  • D ( 1 , 𝑒 ) , ο€Ή 1 , 𝑒  βˆ’ 1
  • Eno horizontal tangents.

Find all points on this curve where the tangent is vertical.

  • A ( 𝑒 , 1 ) , ο€Ή 𝑒 , 1  βˆ’ 1
  • B ( 1 , 𝑒 )
  • C ( 1 , 𝑒 ) , ο€Ή 1 , 𝑒  βˆ’ 1 , ( 𝑒 , 1 ) , ο€Ή 𝑒 , 1  βˆ’ 1
  • D ( 1 , 𝑒 ) , ο€Ή 1 , 𝑒  βˆ’ 1
  • Eno vertical tangents.

Q15:

A cycloid curve is given by the equations π‘₯ = π‘Ÿ ( 𝑑 βˆ’ 𝑑 ) s i n and 𝑦 = π‘Ÿ ( 1 βˆ’ 𝑑 ) c o s .

Find the tangent to the cycloid at the point where 𝑑 = πœ‹ 3 .

  • A 𝑦 βˆ’ π‘Ÿ 2 = √ 3 2 ο€Ώ π‘₯ βˆ’ π‘Ÿ πœ‹ 3 + π‘Ÿ √ 3 2 
  • B 𝑦 βˆ’ π‘Ÿ 2 = 1 √ 3 ο€Ώ π‘₯ βˆ’ π‘Ÿ πœ‹ 3 βˆ’ π‘Ÿ √ 3 2 
  • C 𝑦 βˆ’ π‘Ÿ 2 = 1 2 ο€Ώ π‘₯ βˆ’ π‘Ÿ πœ‹ 3 βˆ’ π‘Ÿ √ 3 2 
  • D 𝑦 βˆ’ π‘Ÿ 2 = √ 3 ο€Ώ π‘₯ βˆ’ π‘Ÿ πœ‹ 3 + π‘Ÿ √ 3 2 
  • E 𝑦 βˆ’ π‘Ÿ 2 = βˆ’ 1 √ 3 ο€Ώ π‘₯ βˆ’ π‘Ÿ πœ‹ 3 βˆ’ π‘Ÿ √ 3 2 

Find all points on the curve where the tangent is horizontal.

  • A ( πœ‹ π‘Ÿ ( 2 𝑛 βˆ’ 1 ) , 2 π‘Ÿ ) where 𝑛 is an integer
  • B ( 2 𝑛 πœ‹ π‘Ÿ , 2 π‘Ÿ ) where 𝑛 is an integer
  • C ( πœ‹ π‘Ÿ ( 2 𝑛 βˆ’ 1 ) , 2 π‘Ÿ ) , ( 2 𝑛 πœ‹ π‘Ÿ , 0 ) where 𝑛 is an integer
  • D ( 2 𝑛 πœ‹ π‘Ÿ , 0 ) where 𝑛 is an integer
  • E ( 2 𝑛 πœ‹ π‘Ÿ , 0 ) , ( πœ‹ π‘Ÿ ( 2 𝑛 βˆ’ 1 ) , 2 π‘Ÿ ) where 𝑛 is an integer

Find all points on the curve where the tangent is vertical.

  • A ( 2 𝑛 πœ‹ π‘Ÿ , 0 ) , ( πœ‹ π‘Ÿ ( 2 𝑛 βˆ’ 1 ) , 2 π‘Ÿ ) where 𝑛 is an integer
  • B ( πœ‹ π‘Ÿ ( 2 𝑛 βˆ’ 1 ) , 2 π‘Ÿ ) where 𝑛 is an integer
  • C ( 2 𝑛 πœ‹ π‘Ÿ , 0 ) where 𝑛 is an integer
  • D ( πœ‹ π‘Ÿ ( 2 𝑛 βˆ’ 1 ) , 2 π‘Ÿ ) , ( 2 𝑛 πœ‹ π‘Ÿ , 0 ) where 𝑛 is an integer
  • E ( 2 𝑛 πœ‹ π‘Ÿ , 2 π‘Ÿ ) where 𝑛 is an integer

Q16:

Find all possible equations of the tangents to the curve π‘₯ = 3 𝑑 + 1 2 , 𝑦 = 2 𝑑 + 1 3 that pass through the point ( 4 , 3 ) .

  • A 𝑦 = π‘₯ βˆ’ 1 , 𝑦 = βˆ’ π‘₯ + 7
  • B 𝑦 = π‘₯ βˆ’ 1
  • C 𝑦 = βˆ’ 2 π‘₯ + 1 1
  • D 𝑦 = π‘₯ βˆ’ 1 , 𝑦 = βˆ’ 2 π‘₯ + 1 1
  • E 𝑦 = π‘₯ βˆ’ 1 , 𝑦 = βˆ’ π‘₯ + 7 , 𝑦 = βˆ’ 2 π‘₯ + 1 1

Q17:

Find the value of π‘š at which the curve π‘₯ = 8 π‘š + 5 π‘š + π‘š βˆ’ 1 3 2 , 𝑦 = 5 π‘š βˆ’ π‘š + 2 2 has a vertical tangent.

  • A 1 4 , 1 6
  • B 1 1 0
  • C 1 4
  • D βˆ’ 1 6 , βˆ’ 1 4
  • E 1 6

Q18:

Determine the equation of the tangent to the curve π‘₯ = 𝑛 + 9 𝑛 2 , 𝑦 = 𝑛 2 at 𝑛 = βˆ’ 6 .

  • A 4 𝑦 βˆ’ π‘₯ βˆ’ 1 6 2 = 0
  • B 𝑦 + 4 π‘₯ + 3 6 = 0
  • C βˆ’ 4 𝑦 βˆ’ π‘₯ + 1 2 6 = 0
  • D 𝑦 βˆ’ 4 π‘₯ βˆ’ 1 0 8 = 0

Q19:

Consider the curve π‘₯ = 𝑑 βˆ’ 3 𝑑 3 , 𝑦 = 𝑑 βˆ’ 3 2 .

Find all points on this curve where the tangent is horizontal.

  • A ( 0 , βˆ’ 3 ) , ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , βˆ’ 2 )
  • B ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , βˆ’ 2 )
  • C ( 0 , βˆ’ 3 ) , ( 2 , βˆ’ 2 )
  • D ( 0 , βˆ’ 3 )
  • EThere are no horizontal tangents.

Find all points on this curve where the tangent is vertical.

  • A ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , βˆ’ 2 )
  • B ( 0 , βˆ’ 3 ) , ( 2 , βˆ’ 2 )
  • C ( 0 , βˆ’ 3 ) , ( 2 , βˆ’ 2 ) , ( βˆ’ 2 , βˆ’ 2 )
  • D ( 0 , βˆ’ 3 )
  • EThere are no vertical tangents.

Q20:

Consider the curve π‘₯ = πœƒ c o s , 𝑦 = 3 πœƒ c o s .

Find all points on this curve where the tangent is horizontal.

  • A ο€Ό βˆ’ 1 2 , 1 
  • B ο€Ό 1 2 , βˆ’ 1 
  • C ( 0 , 0 ) , ο€Ό 1 2 , βˆ’ 1 
  • D ο€Ό 1 2 , βˆ’ 1  , ο€Ό βˆ’ 1 2 , 1 
  • EThere are no horizontal tangents.

Find all points on this curve where the tangent is vertical.

  • AThere are no vertical tangents.
  • B ο€Ό βˆ’ 1 2 , 1 
  • C ο€Ό 1 2 , βˆ’ 1 
  • D ο€Ό 1 2 , βˆ’ 1  , ο€Ό βˆ’ 1 2 , 1 
  • E ( 0 , 0 ) , ο€Ό 1 2 , βˆ’ 1 

Q21:

Find an equation of the tangent to the curve π‘₯ = πœ‹ 𝑑 s i n , 𝑦 = 𝑑 + 𝑑 2 at the point ( 0 , 2 ) .

  • A 𝑦 = βˆ’ 3 πœ‹ π‘₯ + 2
  • B 𝑦 = βˆ’ πœ‹ 3 π‘₯ + 2
  • C 𝑦 = πœ‹ 3 π‘₯ + 2
  • D 𝑦 = βˆ’ 3 πœ‹ π‘₯ + 2
  • E 𝑦 = 3 πœ‹ π‘₯ + 2

Q22:

Find the equation of the normal to the curve π‘₯ = 3 πœƒ c o s , 𝑦 = √ 2 + πœƒ s i n at πœƒ = 3 πœ‹ 4 .

  • A 𝑦 βˆ’ 1 3 π‘₯ βˆ’ 4 √ 2 3 = 0
  • B 𝑦 + 1 3 π‘₯ βˆ’ 5 √ 2 3 = 0
  • C 1 3 𝑦 βˆ’ π‘₯ = 0
  • D βˆ’ 1 3 𝑦 βˆ’ π‘₯ + √ 2 = 0

Q23:

Find the equation of the tangent to the curve π‘₯ = βˆ’ 9 πœƒ c s c , 𝑦 = 4 πœƒ βˆ’ 5 πœƒ c o t c s c at πœƒ = 5 πœ‹ 3 .

  • A βˆ’ 3 π‘₯ + 𝑦 + 1 6 √ 3 = 0
  • B βˆ’ π‘₯ 3 + 𝑦 = 0
  • C 3 π‘₯ + 𝑦 βˆ’ 2 0 √ 3 = 0
  • D π‘₯ 3 + 𝑦 βˆ’ 4 √ 3 = 0

Q24:

Find the equation of the tangent to the curve π‘₯ = 5 πœƒ βˆ’ 2 c o t , 𝑦 = 4 πœƒ + √ 2 πœƒ s i n s e c 2 at the point πœƒ = πœ‹ 4 .

  • A βˆ’ 3 π‘₯ 5 + 𝑦 βˆ’ 2 9 5 = 0
  • B 5 π‘₯ 3 + 𝑦 βˆ’ 9 = 0
  • C π‘₯ + 4 𝑦 βˆ’ 3 5 = 0
  • D 3 π‘₯ 5 + 𝑦 βˆ’ 2 9 5 = 0

Q25:

Find the equation of the tangent to the curve π‘₯ = 9 πœƒ c o s , 𝑦 = √ 2 + 3 πœƒ s i n at πœƒ = 3 πœ‹ 4 .

  • A 1 3 𝑦 βˆ’ π‘₯ βˆ’ √ 2 = 0
  • B 𝑦 βˆ’ 1 3 π‘₯ βˆ’ 5 √ 2 3 = 0
  • C βˆ’ 1 3 𝑦 βˆ’ π‘₯ = 0
  • D 𝑦 + 1 3 π‘₯ βˆ’ 4 √ 2 3 = 0