RD Sharma Solutions For Class 12 Maths Exercise 16.2 Chapter 16 Tangents and Normals

Highly experienced subject experts at BYJU’S, having vast knowledge of concepts prepare the answers, which match the understanding ability of the students. This exercise, available in PDF format, explains the equation of tangent and normal to a given curve. A straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point, is called a tangent. RD Sharma Solutions for Class 12 Chapter 16 Tangents and Normals Exercise 16.2 can be used as their reference material. Some of the important topics of this exercise are listed below.

  • Equations of tangents and normal
  • Finding the equations of tangent and normal to a curve at a given point
  • Finding tangent and normal parallel or perpendicular to a given line
  • Finding tangent or normal passing through a given point
  • Miscellaneous applications of tangents and normals

RD Sharma Solutions For Class 12 Tangents and Normals Exercise 16.2:Download PDF Here

RD Sharma Class 12 Maths Solutions Chapter 16 Tangents And Normals Exercise 16.2
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Access other exercises of RD Sharma Solutions For Class 12 Chapter 16 – Tangents and Normals

Exercise 16.1 Solutions

Exercise 16.3 Solutions

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 16 – Tangents and Normals Exercise 16.2

Exercise 16.2 Page No: 15.27

1. Find the equation of the tangent to the curve √x + √y = a, at the point (a2/4, a2/4).

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 47

2. Find the equation of the normal to y = 2x3 – x2 + 3 at (1, 4).

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 48

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 49

3. Find the equation of the tangent and the normal to the following curves at the indicated points:

(i) y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5)

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 50

(ii) y = x4 – 6x3 + 13x2 – 10x + 5 at x = 1 y = 3

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 51

(iii) y = x2 at (0, 0)

Solution:

Given y = x2 at (0, 0)

By differentiating the given curve, we get the slope of the tangent

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 52

m (tangent) at (x = 0) = 0

Normal is perpendicular to tangent so, m1m2 = – 1

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 53

We can see that the slope of normal is not defined

Equation of tangent is given by y – y1 = m (tangent) (x – x1)

y = 0

Equation of normal is given by y – y1 = m (normal) (x – x1)

x = 0

(iv) y = 2x2 – 3x – 1 at (1, – 2)

Solution:

Given y = 2x2 – 3x – 1 at (1, – 2)

By differentiating the given curve, we get the slope of the tangent

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 54

m (tangent) at (1, – 2) = 1

Normal is perpendicular to tangent so, m1m2 = – 1

m (normal) at (1, – 2) = – 1

Equation of tangent is given by y – y1 = m (tangent) (x – x1)

y + 2 = 1(x – 1)

y = x – 3

Equation of normal is given by y – y1 = m (normal) (x – x1)

y + 2 = – 1(x – 1)

y + x + 1 = 0

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 55

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 56

Equation of tangent is given by y – y1 = m (tangent) (x – x1)

y + 2 = – 2(x – 2)

y + 2x = 2

Equation of normal is given by y – y1 = m (normal) (x – x1)

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 57

2y + 4 = x – 2

2y – x + 6 = 0

4. Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 58

5. Find the equation of the tangent and the normal to the following curves at the indicated points:

(i) x = θ + sin θ, y = 1 + cos θ at θ = π/2

Solution:

Given x = θ + sin θ, y = 1 + cos θ at θ = π/2

By differentiating the given equation with respect to θ, we get the slope of the tangent

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 59

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 60

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 61

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(iii) x = at2, y = 2at at t = 1.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 63

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(iv) x = a sec t, y = b tan t at t.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 65

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 66

(v) x = a (θ + sin θ), y = a (1 – cos θ) at θ

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 67

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 68

(vi) x = 3 cos θ – cos3 θ, y = 3 sin θ – sin3θ

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 69

6. Find the equation of the normal to the curve x2 + 2y2 – 4x – 6y + 8 = 0 at the point whose abscissa is 2.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 70

Finding y co – ordinate by substituting x in the given curve

2y2 – 6y + 4 = 0

y2 – 3y + 2 = 0

y = 2 or y = 1

m (tangent) at x = 2 is 0

Normal is perpendicular to tangent so, m1m2 = – 1

m (normal) at x = 2 is 1/0, which is undefined

Equation of normal is given by y – y1 = m (normal) (x – x1)

x = 2

7. Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3).

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 71

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 72

8. The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.

Solution:

Given y2 = ax3 + b is y = 4x – 5

By differentiating the given curve, we get the slope of the tangent

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 73

m (tangent) at (2, 3) = 2a

Equation of tangent is given by y – y1 = m (tangent) (x – x1)

Now comparing the slope of a tangent with the given equation

2a = 4

a = 2

Now (2, 3) lies on the curve, these points must satisfy

32 = 2 × 23 + b

b = – 7

9. Find the equation of the tangent line to the curve y = x2 + 4x – 16 which is parallel to the line 3x – y + 1 = 0.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 74

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 75

10. Find the equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x + 14y + 4 = 0.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 76

RD Sharma Solutions for Class 12 Maths Chapter 16 Tangents and Normals Image 77

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