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 = .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: 

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

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

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