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Textbook | NCERT |

Class | Class 12th |

Subject | Maths |

Chapter | 16 |

Exercise | 16.1 |

Category | RD Sharma Solutions |

**RD Sharma Class 12 Ex 16.1 Solutions Chapter 16 Tangents and Normals**

**Question 1. Find the slopes of the tangent and the normal to the following curves at the indicated points:**

**(i) y = √x ^{3} at x = 4**

**(ii) y = √x at x = 9**

**(iii) y = x ^{3} – x at x = 2**

**(iv) y = 2x ^{2} + 3 sin x at x = 0**

**(v) x = a(θ – sin θ), y = a(1 + cos θ) at θ = –π/2**

**(vi) x = a cos ^{3} θ, y = a sin^{3} θ at θ = π/4**

**(vii) x = a(θ – sin θ), y = a(1 – cos θ) at θ = π/2**

**(viii) y = (sin 2x + cot x + 2) ^{2} at x = π/2**

**(ix) x ^{2} + 3y + y^{2} = 5 at (1, 1)**

**(x) xy = 6 at (1, 6)**

**Solution:**

We know that the slope of the tangent of a curve is given by dy/dx.

And slope of the normal = –1/slope of tangent = –1/(dy/dx).

**(i) y = √x**^{3} at x = 4

^{3}at x = 4

Differentiating y = √x

^{3}with respect to x, we get,Slope of tangent = 3x

^{1/2}/2At x = 4, slope of tangent becomes,

Slope of tangent = 3(4)

^{1/2}/2 = 3(2)/2 = 3And slope of normal at x = 4 is –1/3.

**(ii) y = √x at x = 9**

Differentiating y = √x with respect to x, we get,

Slope of tangent = x

^{–1/2}/2At x = 9, slope of tangent becomes,

Slope of tangent = 9

^{–1/2}/2 = 1/[(3)(2)] = 1/6And slope of normal at x = 9 is –6.

**(iii) y = x**^{3} – x at x = 2

^{3}– x at x = 2

Differentiating y = x

^{3}– x with respect to x, we get,Slope of tangent = 3x

^{2}– 1At x = 2, slope of tangent becomes,

Slope of tangent = 3(2)

^{2}– 1 = 3(4) – 1 = 11And slope of normal at x = 2 is –1/11.

**(iv) y = 2x**^{2} + 3 sin x at x = 0

^{2}+ 3 sin x at x = 0

Differentiating y = 2x

^{2}+ 3 sin x with respect to x, we get,Slope of tangent = 4x + 3 cos x

At x = 0, slope of tangent becomes,

Slope of tangent = 4(0) + 3 cos 0 = 3.

And slope of normal at x = 0 is –1/3.

**(v) x = a (θ – sin θ), y = a (1 + cos θ) at θ = –π/2**

Differentiating x = a (θ – sin θ) with respect to θ, we get,

=> dx/dθ = a (1 – cos θ) . . . . (1)

Differentiating y = a (1 + cos θ) with respect to θ, we get,

=> dy/dθ = a (–sin θ) . . . . (2)

Dividing (2) by (1), we get,

dy/dx = Slope of tangent = –sin θ/(1 – cos θ)

At θ = –π/2, slope of tangent becomes,

Slope of tangent = –sin (–π/2)/(1 – cos (–π/2))

= 1/(1–0)

= 1

And slope of normal at θ = –π/2 is –1.

**(vi) x = a cos**^{3} θ, y = a sin^{3} θ at θ = π/4

^{3}θ, y = a sin

^{3}θ at θ = π/4

Differentiating x = a cos

^{3}θ with respect to θ, we get,=> dx/dθ = a [(3cos

^{2}θ) (–sin θ)]= –3a cos

^{2}θ sin θ . . . . (1)Differentiating y = a sin

^{3}θ with respect to θ, we get,=> dy/dθ = a [(3sin

^{2}θ) (cos θ)]= 3a sin

^{2}θ cos θ . . . . (2)Dividing (2) by (1), we get,

dy/dx = Slope of tangent == – tan θ

At θ = π/4, slope of tangent becomes,

Slope of tangent = – tan (π/4)

= −1

And slope of normal at θ = π/4 is 1.

**(vii) x = a (θ – sin θ), y = a (1 – cos θ) at θ = π/2**

Differentiating x = a (θ – sin θ) with respect to θ, we get,

=> dx/dθ = a (1 – cos θ) . . . . (1)

Differentiating y = a (1 – cos θ) with respect to θ, we get,

=> dy/dθ = a (sin θ) . . . . (2)

Dividing (2) by (1), we get,

dy/dx = Slope of tangent = sin θ/(1−cosθ)

= – tan θ

At θ = π/2, slope of tangent becomes,

Slope of tangent = sin π/2/(1−cos π/2)

= 1/(1−0)

= 1

And slope of normal at θ = π/2 is −1.

**(viii) y = (sin 2x + cot x + 2)**^{2} at x = π/2

^{2}at x = π/2

Differentiating y = (sin 2x + cot x + 2)

^{2}with respect to x, we get,Slope of tangent = 2 (sin 2x + cot x + 2) (2 cos 2x – cosec

^{2}x)At x = π/2, slope of tangent becomes,

Slope of tangent = 2 (sin 2(π/2) + cot π/2 + 2) (2 cos 2(π/2) – cosec

^{2}(π/2))= 2 (0 + 0 + 2) (–2 – 1)

= –12

And slope of normal at x = π/2 is 1/12.

**(ix) x**^{2} + 3y + y^{2} = 5 at (1, 1)

^{2}+ 3y + y

^{2}= 5 at (1, 1)

Differentiating x

^{2}+ 3y + y^{2}= 5 with respect to x, we get,=> 2x + 3 (dy/dx) + 2y (dy/dx) = 0

=> 2x + dy/dx (2y+3) = 0

=> Slope of tangent = dy/dx = –2x/(2y+3)

At x = 1 and y = 1, slope of tangent becomes,

Slope of tangent = –2(1)/[2(1)+3] = –2/5

And slope of normal at (1, 1) is 5/2.

**(x) xy = 6 at (1, 6)**

Differentiating xy = 6 with respect to x, we get,

=> x (dy/dx) + y = 0

=> Slope of tangent = dy/dx = –y/x

At x = 1 and y = 6, slope of tangent becomes,

Slope of tangent = –6/1 = –6

And slope of normal at (1, 6) is 1/6.

**Question 2. Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2.**

**Solution:**

We know that the slope of the tangent of a curve is given by dy/dx.

Differentiating xy + ax + by = 2 with respect to x, we get

=> x (dy/dx) + y + a + b (dy/dx) = 2

=> dy/dx = −(a+y)/(x+b)

As we are given dy/dx = 2, we get,

=> −(a+y)/(x+b) = 2

Now at x = 1 and y = 1, we get,

=> −(a+1)/(1+b) = 2

=> −a − 1 = 2 + 2b

=> a + 2b = –3 . . . . (1)

Now the point (1, 1) also lies on the curve, so we have,

=> 1 × 1 + a × 1 + b × 1 = 2

=> 1 + a + b = 2

=> a + b = 1 . . . . (2)

Subtracting (1) from (2), we get,

=> –b = 1+3

=> b = –4

Putting b = –4 in (1), we get,

=> a = 1+4 = 5

Therefore, the value of a is 5 and b is –4.

**Question 3. If the tangent to the curve y = x**^{3} + ax + b at (1, –6) is parallel to the line x – y + 5 = 0, find a and b.

^{3}+ ax + b at (1, –6) is parallel to the line x – y + 5 = 0, find a and b.

**Solution:**

We know that the slope of the tangent of a curve is given by dy/dx.

Differentiating y = x

^{3}+ ax + b with respect to x, we get=> dy/dx = 3x

^{2}+ aNow at x = 1 and y = –6, we get,

=> dy/dx = 3(1)

^{2}+ a=> dy/dx = 3 + a . . . . (1)

Now this curve is parallel to the line x – y + 5 = 0.

=> y = x + 5

Therefore slope of the line is 1. So, the slope of the curve will also be 1 as slope of parallel lines are equal. So, from (1), we get,

=> dy/dx = 1 . . . . (2)

From (1) and (2), we get,

=> a + 3 = 1

=> a = –2 . . . . (3)

Now at x = 1 and y = –6, our curve y = x

^{3}+ ax + b becomes,=> –6 = 1 + a + b

=> a + b = –7

Using (3), we get,

=> b = –7 – (–2)

=> b = –5

Therefore, the value of a is –2 and b is –5.

**Question 4. Find a point on the curve y = x**^{3} – 3x where the tangent is parallel to the chord joining (1, – 2) and (2, 2).

^{3}– 3x where the tangent is parallel to the chord joining (1, – 2) and (2, 2).

**Solution:**

We are given the coordinates of the chord (1, – 2) and (2, 2).

Therefore, slope of the chord == 4

Given curve is y = x

^{3}– 3x. We know that the slope of the tangent of a curve is given by dy/dx.=> dy/dx = 3x

^{2}– 3As the tangent is parallel to the chord, its slope must be equal to 4.

=> 3x

^{2}– 3 = 4=> 3x

^{2}= 7=> x =

Putting value of x in the curve y = x

^{3}– 3x, we get=> y = x (x

^{2}– 3)=> y =

=> y =

Therefore, the required point is.

**Question 5. Find a point on the curve y = x**^{3} – 2x^{2} – 2x at which the tangent lines are parallel to the line y = 2x – 3.

^{3}– 2x

^{2}– 2x at which the tangent lines are parallel to the line y = 2x – 3.

**Solution:**

Given curve is y = x

^{3}– 2x^{2 }– 2x. We know that the slope of the tangent of a curve is given by dy/dx.=> dy/dx = 3x

^{2}– 4x – 2 . . . . (1)Now this curve is parallel to the line y = 2x – 3 whose slope is 2. So, the slope of the curve will also be 2. So, from (1), we get,

=> 3x

^{2}– 4x – 2 = 2=> 3x

^{2}– 6x + 2x – 4 = 0=> 3x (x – 2) + 2 (x – 2) = 0

=> (x – 2) (3x + 2) = 0

=> x = 2 or x = –2/3

If x = 2, we get

y = (2)

^{3}– 2 × (2)^{2}– 2 × (2)= 8 – 8 – 4

= – 4

And if x = –2/3, we get,

y = (–2/3)

^{3}– 2 × (–2/3)^{2}– 2 × (–2/3)=

= 4/27

Therefore, (2, –4) and (–2/3, 4/27) are the required points.

**Question 6. Find a point on the curve y**^{2} = 2x^{3} at which the slope of the tangent is 3.

^{2}= 2x

^{3}at which the slope of the tangent is 3.

**Solution:**

Given curve is y

^{2 }= 2x^{3}. We know that the slope of the tangent of a curve is given by dy/dx.=> 2y dy/dx = 6x

^{2}=> dy/dx = 3x

^{2}/y . . . . (1)As it is given that the slope of tangent is 3, we get,

=> 3x

^{2}/y = 3=> 3x

^{2}= 3y=> x

^{2}= yPutting this in the curve y

^{2}= 2x^{3}, we get,=> (x

^{2})^{2}= 2x^{3}=> x

^{4}− 2x^{3}= 0=> x

^{3}(x − 2) = 0=> x = 0 or x = 2

If x = 0, we get, y = 0. Putting these values in (1), we get dy/dx = 0, which is not possible as the given value of slope is 3.

And if x =2, we get y = 4.

Therefore, the required point is (2, 4).

**Question 7. Find a point on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45**^{o} with the x–axis.

^{o}with the x–axis.

**Solution:**

Given curve is xy + 4 = 0. We know that the slope of the tangent of a curve is given by dy/dx.

=> x (dy/dx) + y = 0

=> dy/dx = −y/x . . . . (1)

We are given that the tangent is inclined at an angle of 45

^{o}with the x–axis. So slope of the tangent is,dy/dx = tan 45

^{o}= 1.So, (1) becomes,

=> −y/x = 1

=> y = −x

Putting this in the curve xy + 4 = 0 we get,

=> x(−x) + 4 = 0

=> x

^{2}= 4=> x = ±2

When x = 2, y = −2.

And when x = −2, y = 2.

Therefore, the required points are (2, – 2) & (– 2, 2).

**Question 8. Find a point on the curve y = x**^{2 }where the slope of the tangent is equal to the x – coordinate of the point.

^{2 }where the slope of the tangent is equal to the x – coordinate of the point.

**Solution:**

Given curve is y = x

^{2}. We know that the slope of the tangent of a curve is given by dy/dx.=> dy/dx = 2x . . . . (1)

It is given that the slope of the tangent is equal to the x – coordinate of the point.

Therefore, dy/dx = x . . . . (2)

From (1) & (2), we get,

2x = x

=> x = 0

Putting this in the curve y = x

^{2}, we get,=> y = 0

^{2}=> y = 0

Therefore, the required point is (0, 0).

**Question 9. At what point on the circle x**^{2} + y^{2} – 2x – 4y + 1 = 0, the tangent is parallel to x – axis.

^{2}+ y

^{2}– 2x – 4y + 1 = 0, the tangent is parallel to x – axis.

**Solution:**

Given circle is x

^{2}+ y^{2}– 2x – 4y + 1 = 0. We know that the slope of the tangent of a curve is given by dy/dx.=> 2x + 2y (dy/dx) – 2 – 4 (dy/dx) = 0

=> dy/dx = (1– x)/(y– 2)

As the tangent is parallel to x-axis, its slope is equal to 0.

So, (1– x)/(y– 2) = 0

=> x = 1

Putting x = 1 in the circle x

^{2}+ y^{2}– 2x – 4y + 1 = 0, we get,=> 1 + y

^{2}– 2 – 4y +1 = 0=> y

^{2}– 4y = 0=> y (y – 4) = 0

=> y = 0 and y = 4

Therefore, the required points are (1, 0) and (1, 4).

**Question 10. At what point of the curve y = x**^{2} does the tangent make an angle of 45^{o} with the x–axis?

^{2}does the tangent make an angle of 45

^{o}with the x–axis?

**Solution:**

Given curve is y = x

^{2}. We know that the slope of the tangent of a curve is given by dy/dx.=> dy/dx = 2x . . . . (1)

We are given that the tangent is inclined at an angle of 45

^{o}with the x–axis. So slope of the tangent isTherefore, dy/dx = tan 45

^{o}= 1 . . . . (2)From (1) & (2), we get,

2x = 1

=> x = 1/2

Putting this in the curve y = x

^{2}, we get,=> y = (1/2)

^{2}=> y = 1/4

Therefore, the required point is (1/2, 1/4).Question 11. Find the points on the curve y = 3x^{2}− 9x + 8 at which the tangents are equally inclined with the axes.Solution:

Given curve is y = 3x^{2}− 9x + 8. We know that the slope of the tangent of a curve is given by dy/dx.

=> dy/dx = 6x − 9 . . . . (1)

We are given that the tangent is equally inclined with the axes. So θ = π/4 or –π/4.

Hence, slope of the tangent is ±1.

=> 6x − 9 = 1 or 6x − 9 = –1

=> 6x = 10 or 6x = 8

=> x = 5/3 or x = 4/3

When x = 5/3,

y = 3 (5/3)^{2}− 9 (5/3) + 8 = 4/3

When x = 4/3,

y = 3 (4/3)^{2}− 9 (5/3) + 8 = 4/3Therefore, the required points are (5/3, 4/3) and (4/3, 4/3).Question 12. At what points on the curve y = 2x^{2}− x + 1 is the tangent parallel to the line y = 3x + 4?Solution:

Given curve is y = 2x^{2}− x + 1. We know that the slope of the tangent of a curve is given by dy/dx.

=> dy/dx = 4x − 1 . . . . (1)

We are given that the tangent is parallel to the line y = 3x + 4. Now the slope of the line is 3, so slope of tangent must also be 3. So, we have,

=> 4x − 1 = 3

=> x = 1

Putting x = 1 in the curve y = 2x^{2}− x + 1, we get

y = 2(1) − 1 + 1 = 2Therefore, the required point is (1, 2).Question 13. Find the point on the curve y = 3x^{2}+ 4 at which the tangent is perpendicular to the line whose slope is −1/6.Solution:

Given curve is y = 3x^{2}+ 4. We know that the slope of the tangent of a curve is given by dy/dx.

=> dy/dx = 6x

It is given that the tangent is perpendicular to the line whose slope is −1/6. So the product of both the slopes must be −1.

Therefore the slope of tangent, dy/dx = 6.

=> 6x = 6

=> x = 1

Putting x = 1 in the curve y = 3x^{2}+ 4, we get,

=> y = 3(1)^{2}+ 4 = 3 + 4 = 7Therefore, (1, 7) is the required point.Question 14. Find the points on the curve x^{2}+ y^{2}= 13, the tangent at each one of which is parallel to the line 2x + 3y = 7.Solution:

Given curve is x^{2}+ y^{2}= 13. We know that the slope of the tangent of a curve is given by dy/dx.

=> 2x + 2y dy/dx = 0

=> dy/dx = −x/y . . . . (1)

It is given that the tangent is parallel to the line 2x + 3y = 7.

=> 3y = −2x + 7

=> y = −(2/3)x + 7/3

Therefore slope of the line is −2/3 and slope of the tangent is also −2/3 as slope of parallel lines are equal.

=> dy/dx = −2/3 . . . . (2)

From (1) and (2), we get,

=> −x/y = −2/3

=> x = 2y/3 . . . . (3)

Putting x = 2y/3 in the curve x^{2}+ y^{2}= 13, we get,

=> 4y^{2}/9 + y^{2}= 13

=> 13y^{2}/9 = 13

=> y^{2}= 9

=> y = ±3

Putting y = ±3, in (3), we get,

When y = 3, x = 2 and when y = −3, x = −2.Therefore, the required points are (2, 3) and (−2, −3).Question 15. Find the points on the curve 2a^{2}y = x^{3}− 3ax^{2}where the tangent is parallel to x-axis.Solution:

Given curve is 2a^{2}y = x^{3}− 3ax^{2}. We know that the slope of the tangent of a curve is given by dy/dx.

=> 2a^{2}dy/dx = 3x^{2}− 3a (2x)

=> dy/dx =

It is given that the tangent is parallel to x-axis, so the slope of the tangent becomes 0.

=> = 0

=> 3x (x − 2a) = 0

=> x = 0 or x = 2a

When x = 0, the value of y from the curve is,

=> y =

=> y =

=> y = 0

And when x = 2a, the value of y is,

=> y =

=> y =

=> y = −2aTherefore, the required points are (0, 0) and (2a, −2a).Question 16. At what points on the curve y = x^{2 }− 4x + 5 is the tangent perpendicular to the line 2y + x = 7?Solution:

Given curve is y = x^{2}− 4x + 5. We know that the slope of the tangent of a curve is given by dy/dx.

=> dy/dx = 2x − 4 . . . . (1)

It is given that the tangent is perpendicular to the line 2y + x = 7.

=> 2y = −x + 7

=> y = −(1/2)x + 7/2

Therefore slope of the line is −1/2 and product of this slope with that of tangent is −1 as both lines are perpendicular to each other.

So, slope of tangent is 2.

=> dy/dx = 2 . . . . (2)

From (1) and (2), we get,

=> 2x − 4 = 2

=> x = 3

Putting this in the curve y = x^{2}− 4x + 5, we get

=> y = x2 − 4x + 5

= (3)^{2}− 4(3) + 5

= 2Therefore, the required point is (3, 2).Question 17. Find points on the curve x^{2}/4 + y^{2}/25 = 1 at which the tangents are(i) parallel to the x-axisSolution:

Given curve is x^{2}/4 + y^{2}/25 = 1. We know that the slope of the tangent of a curve is given by dy/dx.

=> 2x/4 + 2y/25 (dy/dx) = 0

=> dy/dx = −25x/4y

As it is given that the tangent is parallel to the x-axis, its slope must be 0.

=> −25x/4y = 0

=> x = 0

Putting this in the curve x^{2}/4 + y^{2}/25 = 1, we get

=> y^{2}= 25

=> y = ±5Therefore, the required points are (0, 5) and 0, −5).(ii) parallel to the y-axisSolution:

Slope of the tangent = dy/dx = −25x/4y

Therefore, slope of the normal = = 4y/25x

As it is given that the tangent is parallel to the y-axis, the slope of the normal must be 0.

=> 4y/25x = 0

=> y = 0

Putting this in the curve x^{2}/4 + y^{2}/25 = 1, we get

=> x^{2}= 4

=> x = ±2Therefore, the required points are (2, 0) and (−2, 0).Question 18. Find the points on the curve x^{2}+ y^{2}− 2x − 3 = 0 at which the tangents are parallel to(i) x-axisSolution:

Given curve is x^{2 }+ y^{2 }− 2x − 3 = 0. We know that the slope of the tangent of a curve is given by dy/dx.

=> 2x + 2y (dy/dx) − 2 = 0

=> dy/dx = (1−x)/y

As it is given that the tangent is parallel to the x-axis, its slope must be 0.

=> (1−x)/y = 0

=> x = 1

Putting this in the curve x^{2}+ y^{2}− 2x − 3 = 0, we get

=> 1 + y^{2}− 2 − 3 = 0

=> y^{2}= 4

=> y = ±2Therefore, the required points are (1, 2) and (1, −2).(ii) y-axisSolution:

Slope of the tangent = dy/dx = (1−x)/y

Therefore, slope of the normal = = y/(x−1)

As it is given that the tangent is parallel to the y-axis, the slope of the normal must be 0.

=> y/(x−1) = 0

=> y = 0

Putting this in the curve x^{2}+ y^{2}− 2x − 3 = 0, we get

=> x^{2 }− 2x − 3 = 0

=> x = −1, 3Therefore, the required points are (−1, 0) and (3, 0).Question 19. Find points on the curve x^{2}/9 + y^{2}/16 = 1at which the tangents are(i) parallel to the x-axisSolution:

Given curve is x^{2}/9 + y^{2}/16 = 1. We know that the slope of the tangent of a curve is given by dy/dx.

=> 2x/9 + 2y/16 (dy/dx) = 0

=> dy/dx = −16x/9y

As it is given that the tangent is parallel to the x-axis, its slope must be 0.

=> −16x/9y = 0

=> x = 0

Putting this in the curve x^{2}/9 + y^{2}/16 = 1, we get

=> y^{2}= 16

=> y = ±4Therefore, the required points are (0, 4) and 0, −4).(ii) parallel to the y-axisSolution:

Slope of the tangent = dy/dx = −16x/9y

Therefore, slope of the normal = = 9y/16x

As it is given that the tangent is parallel to the y-axis, the slope of the normal must be 0.

=> 9y/16x = 0

=> y = 0

Putting this in the curve x^{2}/9 + y^{2}/16 = 1, we get

=> x^{2}= 9

=> x = ±3Therefore, the required points are (3, 0) and (−3, 0).Question 20. Show that the tangents to the curve y = 7x^{3}+ 11 at the points where x = 2 and x = −2 are parallel.Solution:

Given curve is y = 7x^{3}+ 11. We know that the slope of the tangent of a curve is given by dy/dx.

=> dy/dx = 21x^{2}

Now slope at x = 2 is

=> dy/dx = 21(2)^{2}= 84

And slope at x = −2 is,

=> dy/dx = 21(−2)^{2}= 84

As the slopes at x = 2 and x = −2 are equal, these tangents are parallel.Hence proved.Question 21. Find the points on the curve y = x^{3}where the slope of the tangent is equal to the x-coordinate of the point.Solution:

Given curve is y = x^{3}. We know that the slope of the tangent of a curve is given by dy/dx.

=> dy/dx = 3x^{2}

It is given that the slope of the tangent is equal to the x-coordinate of the point.

=> 3x^{2}= x

=> x(3x − 1) = 0

=> x = 0 or x = 1/3

When x = 0, y = 0^{3}= 0

And when x = 1/3, y = (1/3)^{3}= 1/27Therefore, the required points are (0, 0) and (1/3, 1/27).

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