Chapter 16 Tangents and Normals Ex. 16.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Find the slopes of the tangent and the normal to the curve x = a(θ – sinθ), y =a(1 + cos θ) at θ = begin mathsize 12px style negative straight pi over 2 end style.Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Chapter 16 Tangents and Normals Ex. 16.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Find the equations of the tangent and normal to the given curves at the indicated points:

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

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 3(x)

Solution 3(x)

Question 3(xi)

Solution 3(xi)

Question 3(xii)

Solution 3(xii)

Question 3(xiii)

Solution 3(xiii)

Question 3(xiv)

Solution 3(xiv)

Question 3(xv)

Solution 3(xv)

Question 3(xvi)

Find the equation of the normal to curve y2 = 4x at the point (1, 2) and also find the tangent.Solution 3(xvi)

The equation of the given curve is y2 = 4x . Differentiating with respect to x, we have: 

begin mathsize 12px style 2 straight y dy over dx equals 4 rightwards double arrow dy over dx equals fraction numerator 4 over denominator 2 straight y end fraction equals 2 over straight y therefore right enclose dy over dx end enclose subscript open parentheses 1 comma space 2 close parentheses end subscript equals 2 over 2 equals 1 Now comma space the space slope space at space point space left parenthesis 1 comma space 2 right parenthesis space is space 1 over right enclose begin display style dy over dx end style end enclose subscript open parentheses 1 comma space 2 close parentheses end subscript equals fraction numerator negative 1 over denominator 1 end fraction equals negative 1. therefore Equation space of space the space tangent space at space left parenthesis 1 comma space 2 right parenthesis space is space straight y minus 2 space equals space minus 1 left parenthesis straight x minus 1 right parenthesis. rightwards double arrow straight y minus 2 equals negative straight x plus 1 rightwards double arrow straight x plus straight y minus 3 equals 0 Equation space of space the space normal space is comma straight y minus 2 equals negative left parenthesis negative 1 right parenthesis left parenthesis straight x minus 1 right parenthesis straight y minus 2 equals straight x minus 1 straight x minus straight y plus 1 equals 0 end style

Question 3(xix)

Find the equations of the tangent and the normal to the given curves at the indicated points:

Solution 3(xix)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

From (A)

Equation of tangent is

begin mathsize 12px style open parentheses straight y minus straight a over 5 close parentheses equals 13 over 16 open parentheses straight x minus fraction numerator 2 straight a over denominator 5 end fraction close parentheses 16 straight y minus fraction numerator 16 straight a over denominator 5 end fraction equals 13 straight x minus fraction numerator 26 straight a over denominator 5 end fraction 13 straight x minus 16 straight y minus 2 straight a equals 0 Equation space of space normal space is comma open parentheses straight y minus straight a over 5 close parentheses equals 16 over 13 open parentheses straight x minus fraction numerator 2 straight a over denominator 5 end fraction close parentheses 13 straight y minus fraction numerator 13 straight a over denominator 5 end fraction equals negative 16 straight x plus fraction numerator 32 straight a over denominator 5 end fraction 16 straight x plus 13 straight y minus 9 straight a equals 0 end style

Question 5(iii)

Solution 5(iii)

Question 5(iv)

Solution 5(iv)

Question 5(v)

Solution 5(v)

Question 5(vi)

Find the equations of the tangent and the normal to the following curves at the indicated points:

X = 3 cosθ – cos3θ , y = 3 sinθ – sin3θSolution 5(vi)

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 21

Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0).Solution 21

Question 3(xvii)

Find the equations of the tangent and the normal to the following curves at the indicated points:

Solution 3(xvii)

Given equation curve is 

Differentiating w.r.t x, we get

Slope of tangent at   is

Slope of normal will be

Equation of tangent at   will be

Equation of normal at   is

Question 3(xviii)

Find the equations of the tangent and the normal to the following curves at the indicated points:

Solution 3(xviii)

Given equation curve is 

Differentiating w.r.t x, we get

Slope of tangent at   is

Slope of normal will be

Equation of tangent at   will be

Equation of normal at   is

Question 20

At what points will tangents to the curve   be parallel to x-axis? Also, find the equations of the tangents to the curve at these points.Solution 20

Given equation curve is 

Differentiating w.r.t x, we get

As tangent is parallel to x-axis, its slope will be m = 0

As this point lies on the curve, we can find y

Or

So, the points are (3, 6) and (2, 7).

Equation of tangent at (3, 6) is

y – 6 = 0 (x – 3)

y – 6 = 0

Equation of tangent at (2, 7) is

y – 7 = 0 (x – 2)

y – 7 = 0

Chapter 16 Tangents and Normals Ex. 16.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Find the angle of intersection of the folloing curves

begin mathsize 12px style straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space straight x squared space plus space straight y squared space equals space ab end style

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1 (ix)

Find the angle of intersection of the following curves:

Y = 4 – x2 and y = x2Solution 1 (ix)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.Solution 6

Question 7

Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 9

Solution 9

Question 10

Solution 10

Discover more from EduGrown School

Subscribe to get the latest posts sent to your email.