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# Class 12 RD Sharma Solutions – Chapter 16 Tangents and Normals – Exercise 16.2 | Set 1

• Last Updated : 14 Jul, 2021

### Question 1. Find the equation of the tangent to the curve âˆšx + âˆšy = a at the point (a2/4, a2/4).

Solution:

We have,

âˆšx + âˆšy = a

On differentiating both sides w.r.t. x, we get

dy/dx = -âˆšy/âˆšx

Given, (x1, y1) = (a2/4, a2/4),

Slope of tangent, m =

The equation of tangent is,

y â€“ y1 = m (x â€“ x1)

y â€“ a2/4 = â€“1(x â€“ a2/4)

y â€“ a2/4 = â€“x + a2/4

x + y = a2/2

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

Solution:

We have,

y = 2x3 âˆ’ x2 + 3

On differentiating both sides w.r.t. x, we get

dy/dx = 6x2 – 2x

Slope of tangent =  = 6 (1)2 â€“ 2 (1) = 4

Slope of normal = – 1/Slope of tangent = – 1/4

Given, (x1, y1) = (1, 4),

The equation of normal is,

y – y1 = m (x – x1)

y – 4 = -1/4 (x – 1)

4y – 16 = – x + 1

x + 4y = 17

### (i) y = x4 âˆ’ bx3 + 13x2 âˆ’ 10x + 5 at (0, 5)

Solution:

We have,

y = x4 âˆ’ bx3 + 13x2 âˆ’ 10x + 5

On differentiating both sides w.r.t. x, we get

dy/dx = 4x3 – 3bx2 + 26x – 10

Slope of tangent, m=  = -10

Given, (x1, y1) = (0, 5)

The equation of tangent is,

y – y1 = m (x – x1)

y – 5 = – 10 (x – 0)

y – 5 = -10x

y + 10x – 5 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 5 = 1/10 (x – 0)

10y – 50 = x

x – 10y + 50 = 0

### (ii) y = x4 âˆ’ 6x3 + 13x2 âˆ’ 10x + 5 at x = 1

Solution:

We have,

y = x4 âˆ’ 6x3 + 13x2 âˆ’ 10x + 5

When x = 1, we have y = 1 – 6 + 13 – 10 + 5 = 3

So, (x1, y1) = (1, 3)

Now, y = x4 âˆ’ 6x3 + 13x2 âˆ’ 10x + 5

On differentiating both sides w.r.t. x, we get

dy/dx = 4 x3 – 18 x2 + 26x – 10

Slope of tangent, m =  = 4 – 18 + 26 – 10 = 2

The equation of tangent is,

y – y1 = 2 (x – x1)

y – 3 = 2 (x – 1)

y – 3 = 2x – 2

2x – y + 1 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 3 = -1/2 (x – 1)

2y – 6 = – x + 1

x + 2y – 7 = 0

### (iii) y = x2 at (0, 0)

Solution:

We have,

y = x2

On differentiating both sides w.r.t. x, we get

dy/dx = 2x

Given, (x1, y1) = (0, 0)

Slope of tangent, m=  = 2 (0) = 0

The equation of tangent is,

y – y1 = m (x – x1)

y – 0 = 0 (x – 0)

y = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 0 = -1/0 (x – 0)

x = 0

### (iv) y = 2x2 âˆ’ 3x âˆ’ 1 at (1, âˆ’2)

Solution:

We have,

y = 2x2 âˆ’ 3x âˆ’ 1

On differentiating both sides w.r.t. x, we get

dy/dx = 4x – 3

Given, (x1, y1) = (1, -2)

Slope of tangent, m =  = 4 – 3 = 1

The equation of tangent is,

y – y1 = m (x – x1)

y + 2 = 1 (x – 1)

y + 2 = x – 1

x – y – 3 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y + 2 = -1 (x – 1)

y + 2 = -x + 1

x + y + 1 = 0

### (v)  at (2, -2)

Solution:

We have,

On differentiating both sides w.r.t. x, we get

Given, (x1, y1) = (2, -2)

Slope of tangent, m=  =  = -2

The equation of tangent is,

y – y1 = m (x – x1)

y + 2 = -2 (x – 2)

y + 2 = -2x + 4

2x + y – 2 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y + 2 = 1/2 (x – 2)

2y + 4 = x – 2

x – 2y – 6 = 0

### (vi) y = x2 + 4x + 1 at x = 3

Solution:

We have,

y = x2 + 4x + 1

On differentiating both sides w.r.t. x, we get,

dy/dx = 2x + 4

When x = 3, y = 9 + 12 + 1 = 22

So, (x1, y1) = (3, 22)

Slope of tangent, m=  = 10

The equation of tangent is,

y – y1 = m (x – x1)

y – 22 = 10 (x – 3)

y – 22 = 10x – 30

10x – y – 8 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 22 = -1/10 (x – 3)

10y – 220 = – x + 3

x + 10y – 223 = 0

### (vii)  at (a cos Î¸, b sin Î¸)

Solution:

We have,

On differentiating both sides w.r.t. x, we get

dy/dx = -x b2/y a2

Slope of tangent, m=

Given, (x1, y1) = (a cos Î¸, b sin Î¸)

The equation of tangent is,

y – y1 = m (x – x1)

y – b sin Î¸ = -bcosÎ¸/asinÎ¸  (x – a cos Î¸)

ay sin Î¸ – ab sin2 Î¸ = -bx cos Î¸ + ab cos2 Î¸

bx cos Î¸ + ay sin Î¸ = ab

On dividing by ab, we get

x/a cosÎ¸ + y/b sinÎ¸ = 1

The equation of normal is,

y – y1 = -1/m (x – x1)

y – b sin Î¸ = asinÎ¸/bcosÎ¸  (x – a cos Î¸)

by cos Î¸ – b2 sin Î¸ cos Î¸ = ax sin Î¸ – a2 sin Î¸ cos Î¸

ax sin Î¸ – by cos Î¸ = (a2 – b2) sin Î¸ cos Î¸

On dividing by sin Î¸ cos Î¸, we get

ax sec Î¸ – by cosec Î¸ = a2 – b2

### (viii)  at (a sec Î¸, b tan Î¸)

Solution:

We have,

On differentiating both sides w.r.t. x, we get

dy/dx = x b2/y a2

Slope of tangent}, m=

Given, (x1, y1) = (a sec Î¸, b tan Î¸)

The equation of tangent is,

y – y1 = m (x – x1)

y – b tan Î¸ =  (x – a sec Î¸)

ay sin Î¸ – ab(sin2 Î¸/cos Î¸) = bx – (ab/cos Î¸)

ay sin Î¸ cos Î¸ – ab sin2 Î¸ = bx cos Î¸ – ab

bx cos Î¸ – ay sin Î¸ cos Î¸ = ab (1 – sin2 Î¸)

bx cos Î¸ – ay sin Î¸ cos Î¸ = ab cos2 Î¸

On dividing by ab cos2 Î¸, we get

x/a sec Î¸ – y/b tan Î¸ = 1

The equation of normal is,

y – y1 = -1/m (x – x1)

y – b tan Î¸ = -a sin Î¸/b (x – a sec Î¸)

by – b2 tan Î¸ = -ax sin Î¸ + a2 tan Î¸

ax sin Î¸ + by = (a2 + b2) tan Î¸

On dividing by tan Î¸, we get

ax cos Î¸ + by cot Î¸ = a2 + b2

### (ix) y2 = 4ax at (a/m2, 2a/m)

Solution:

We have,

y2 = 4ax

On differentiating both sides w.r.t. x, we get

dy/dx = 2a/y

Given, (x1, y1) = (a/m2, 2a/m)

Slope of tangent =  = m

The equation of tangent is,

y – y1 = m (x – x1)

my – 2a = m2 x – a

m2 x – my + a = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

m3 y – 2a m2 = – m2 x + a

m2 x + m3 y – 2a m2 – a = 0

### (x) c2 (x2 + y2) = x2y2 at (c/cos Î¸, c/sin Î¸)

Solution:

We have,

c2 (x2 + y2) = x2y2

On differentiating both sides w.r.t. x, we get

2x c2 + 2y c2(dy/dx) = x2 2y(dy/dx) + 2x y2

dy/dx(2y c2 – 2 x2 y) = 2x y2 – 2x c2

Slope of tangent, m=

= -cos3 Î¸/ sin3 Î¸

Given, (x1, y1) = (c/cos Î¸, c/sin Î¸)

The equation of tangent is,

y – y1 = m (x – x1)

sin2 Î¸ (y sin Î¸ – c) = -cos2 Î¸ (x cos Î¸ – c)

y sin3 Î¸ – c sin2 Î¸ = – x cos3 Î¸ + c cos2 Î¸

x cos3 Î¸ + y sin3 Î¸ = c ( sin2 Î¸ + cos2 Î¸)

x cos3 Î¸ + y sin3 Î¸ = c

The equation of normal is,

y – y1 = -1/m (x – x1)

sin3 Î¸ – ycos3 Î¸ = 2c[-cos (2Î¸)/sin(2Î¸)]

sin3 Î¸ – y cos3 Î¸ = -2c cot 2Î¸

sin3 Î¸ – y cos3 Î¸ + 2c cot 2Î¸ = 0

### (xi) xy = c2 at (ct, c/t)

Solution:

We have,

xy = c2

On differentiating both sides w.r.t. x, we get

dy/dx = – y/x

Given, (x1, y1) = (ct, c/t)

Slope of tangent, m=

The equation of tangent is,

y – y1 = m (x – x1)

yt2 – ct = -x + ct

x + y t2 = 2ct

The equation of normal is,

y – y1 = -1/m (x – x1)

y – c/t – t2(x – ct)

yt – c = t3 x – c t4

x t3 – yt = c t4 – c

### (xii) at (x1, y1)

Solution:

We have,

On differentiating both sides w.r.t. x,

dy/dx = – x b2/y a2

Slope of tangent, m=

The equation of tangent is,

y – y1 = m (x – x1)

y – y1 = – x1 b2/y1 a2(x – x1)

y y1 a2 – y12 a2 = -x x1 b2 + x12 b2

x x1 b2 + y y1 a2 = x12 b2 + y12 a2  . . . . (1)

Given (x1, y1) lies on the curve, we get

x12 b2 + y12 a2 = a2 b2

Substituting this in (1), we get

x x1 b2 + y y1 a2 = a2 b2

On dividing this by a2 b2, we get

The equation of normal is,

y – y1 = m (x – x1)

y – y1 = y1 a2/x1 b2(x – x1)

y x1 b2 – x1 y1 b2 = x y1 a2 – x1 y1 a2

x y1 a2 – y x1 b2 = x1 y1 a2 – x1 y1 b2

x y1 a2 – y x1 b2 = x1 y1 (a2 – b2)

On dividing by x1 y1, we get

### (xiii) at (x0 , y0)

Solution:

We have,

On differentiating both sides w.r.t. x, we get

dy/dx = x b2/y a2

Slope of tangent, m=

The equation of tangent is,

y – y1 = m (x – x1)

y – y0 = x0 b2/y0 a2(x – x0)

y y0 a2 – y02 a2 = x x0 b2 – x02 b2

x x0 b2 – y y0 a2 = x02 b2 – y02 a2  . . . . (1)

x02 b2 – y02 a2 = a2 b

Substituting this in eq(1), we get,

x x0 b2 – y y0 a2 = a2 b2

Dividing this by a2 b2, we get

The equation of normal is,

y – y1 = m (x – x1)

y – y0 = y0 a2/x0 b2(x – x0)

y x0 b2 – x0 y0 b2 = -x y0 a2 + x0 y0 a2

x y0 a2 + y x0 b2 = x0 y0 a2 + x0 y0 b2

x y0 a2 + y x0 b2 = x0 y0 (a2 + b2)

Dividing by x0 y0, we get

### (xiv) at (1, 1)

Solution:

We have,

On differentiating both sides w.r.t. x, we get

Slope of tangent, m= = -1

Given, (x1, y1) = (1, 1)

The equation of tangent is,

y – y1 = m (x – x1)

y – 1 = -1 (x – 1)

y – 1 = -x + 1

x + y – 2 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 1 = 1 (x – 1)

y – 1 = x – 1

y – x = 0

### (xv) x2 = 4y at (2, 1)

Solution:

We have,

x2 = 4y

On differentiating both sides w.r.t. x, we get

2x = 4dy/dx

dy/dx = x/2

Slope of tangent, m=  = 2/2 = 1

Given, (x1, y1) = (2, 1)

The equation of tangent is,

y – y1 = m (x – x1)

y – 1 = 1 (x – 2)

y – 1 = x – 2

x – y – 1 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 1 = – 1 (x – 2)

y – 1 = – x + 2

x + y – 3 = 0

### (xvi) y2 = 4x at (1, 2)

Solution:

We have,

y2 = 4x

On differentiating both sides w.r.t. x, we get

2y (dy/dx) = 4

dy/dx = 2/y

Slope of tangent, m= = 2/2 = 1

Given, (x1, y1) = (1, 2)

The equation of tangent is,

y – y1 = m (x – x1)

y – 2 = 1 (x – 1)

y – 2 = x – 1

x – y + 1 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 2 = -1 (x – 1)

y – 2 = -x + 1

x + y – 3 = 0

### (xvii) 4x2 + 9y2 = 36 at (3 cos Î¸, 2 sin Î¸)

Solution:

We have,

4x2 + 9y2 = 36

On differentiating both sides w.r.t. x, we get

8x + 18y dy/dx = 0

18y dy/dx = – 8x

dy/dx = -8x/18y = -4x/9y

Slope of tangent, m =

The equation of tangent is,

y – y1 = m (x – x1)

y – 2 sin Î¸ = -2 cos Î¸/3 sin Î¸(x – 3 cos Î¸)

3y sin Î¸ – 6 sin2 Î¸ = -2x cos Î¸ + 6 cos2 Î¸

2x cos Î¸ + 3y sin Î¸ = 6 (cos2 Î¸ + sin2 Î¸)

2x cos Î¸ + 3y sin Î¸ = 6

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 2 sin Î¸ = -3 sin Î¸/2 cos Î¸(x – 3 cos Î¸)

2y cos Î¸ – 4 sin Î¸ cos Î¸ = 3x sin Î¸ – 9 sin Î¸ cos Î¸

3x sin Î¸ – 2y cos Î¸ – 5sin Î¸ cos Î¸ = 0

### (xviii) y2 = 4ax at (x1, y1)

Solution:

We have,

y2 = 4ax

On differentiating both sides w.r.t. x, we get

2y dy/dx = 4a

dy/dx = 2a/y

At (x1, y1), we have

Slope of tangent = = m

The equation of tangent is,

y – y1 = m (x – x1)

y y1 – y12 = 2ax – 2a x1

y y1 – 4a x1 = 2ax – 2a x1

y y1 = 2ax + 2a x1

y y1 = 2a (x + x1)

The equation of normal is,

y – y1 = -1/m (x – x1)

y – y1 = -y1/2a (x – x1)

### (xix) at (âˆš2a, b)

Solution:

We have,

On differentiating both sides w.r.t. x, we get

dy/dx = x b2/y a2

Slope of tangent, m=

The equation of tangent is,

y – y1 = m (x – x1)

y – b = âˆš2b/a(x – âˆš2a)

ay – ab = âˆš2 bx – 2ab

âˆš2 bx – ay = ab

The equation of normal is,

y – y1 = -1/m (x – x1)

y – b = – a/âˆš2b(x – âˆš2a)

âˆš2 by – âˆš2 b2 = – ax + âˆš2 a

ax + âˆš2 by = âˆš2 b2 + âˆš2 a2

ax/âˆš2 + by = a2 + b2

### Question 4. Find the equation of the tangent to the curve x = Î¸ + sin Î¸, y = 1 + cos Î¸ at Î¸ = Ï€/4.

Solution:

We have,

x = Î¸ + sin Î¸, y = 1 + cos Î¸

and

Slope of tangent, m =

= 1 – âˆš2

Given, (x1, y1) = (Ï€/4 + sin Ï€/4, 1 + cos Ï€/4) = (Ï€/4 + 1/âˆš2, 1 + 1/âˆš2)

The equation of tangent is,

y – y1 = m (x – x1)

y – (1 + 1/âˆš2) = (1 – âˆš2) [x – (Ï€/4 + 1/âˆš2)]

y – 1 – 1/âˆš2 = (1 – âˆš2) (x – Ï€/4 – 1/âˆš2)

### (i) x = Î¸ + sin Î¸, y = 1 + cos Î¸ at Î¸ = Ï€/2

Solution:

We have,

x = Î¸ + sin Î¸ and y = 1 + cos Î¸

dx/dÎ¸ = 1 + cos Î¸ and dy/dÎ¸ = -sinÎ¸

Slope of tangent, m=

= -1/(1 + 0)

= -1

Given, (x1, y1) = (Ï€/2 + sin Ï€/2, 1 + cos Ï€/2) = (Ï€/2 + 1, 1)

The equation of tangent is,

y – y1 = m (x – x1)

y – 1 = -1 (x – Ï€/2 – 1)

2y – 2 = – 2x + Ï€ + 2

x + 2y – Ï€ – 4 = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

y – 1 = 1 (x – Ï€/2 -1)

2y – 2 = 2x – Ï€ – 2

2x – 2y = Ï€

### (ii)  at t = 1/2

Solution:

We have,

dx/dt =

dy/dt =

=

Slope of tangent, m=

Given, (x1, y1) =

The equation of tangent is,

y – y1 = m (x – x1)

80y – 16a = 65x – 26a

65x – 80y – 10a = 0

13x – 16y – 2a = 0

The equation of normal is,

y – y1 = -1/m (x – x1)

65y – 13a = – 80x + 32a

80x + 65y – 45a = 0

16x + 13y – 9a = 0

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

Solution:

We have,

x = at2, y = 2at

dx/dt = 2at and dy/dt = 2a

Slope of tangent, m=  = 1

Given, (x1 , y1) = (a, 2a)

The equation of tangent is,

y – y1 = m (x – x1)

y – 2a = 1 (x – a)

y – 2a = x – a

x – y + a = 0

Equation of normal:

y – y1 = -1/m (x – x1)

y – 2a = – 1 (x – a)

y – 2a = – x + a

x + y = 3a

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

Solution:

We have,

x = a sec t, y = b tan t

dx/dt = a sect tant and dy/dt = b sec2t

Slope of tangent, m =

Given (x1, y1) = (a sec t, b tan t)

The equation of tangent is,

y – y1 = m (x – x1)

y – b tan t = (b/a) cosec t (x – a sec t)

ay sin t cos t – ab sin2 t = bx cos t – ab

bx cos t – ay sin t cos t – ab (1 – sin2 t) = 0

bx cos t – ay sin t cos t = ab cos2 t

On dividing by cos2 t, we get

bx sec t – ay tan t = ab

The equation of normal is,

y – y1 = -1/m (x – x1)

y – b tant = -a/b sint(x – asect)

ycost âˆ’ bsint = âˆ’ a/bâ€‹sint(xcost âˆ’ a)

by cos t – b2 sin t = – ax sin t cos t + a2 sin t

ax sin t cos t + by cos t = (a2 + b2) sin t

On dividing both sides by sin t, we get

ax cos t + by cot t = a2 + b2

### (v) x = a(Î¸ + sin Î¸), y = a(1 âˆ’ cos Î¸) at Î¸

Solution:

We have,

x = a(Î¸ + sin Î¸), y = a(1 âˆ’ cos Î¸)

dx/dÎ¸ = a(1 + cosÎ¸) and dy/dÎ¸ = asinÎ¸

= tan Î¸/2   . . . . (1)

Slope of tangent, m=

Given (x1, y1) = [a(Î¸ + sin Î¸), a(1 âˆ’ cos Î¸)]

The equation of tangent is,

y – y1 = m (x – x1)

y – a (1 – cos Î¸) = tan Î¸/2 [x – a (Î¸ + sin Î¸)]

y âˆ’ a(2sin2Î¸/2â€‹) = xtanÎ¸/2â€‹ âˆ’ aÎ¸tanÎ¸/2 â€‹âˆ’ atanÎ¸/2â€‹sinÎ¸

y âˆ’ 2asin2Î¸/2 â€‹ =(x âˆ’ aÎ¸)tan Î¸/2 âˆ’ 2asin2Î¸/2â€‹

y = (x – aÎ¸) tan Î¸/2

The equation of normal is,

y – a (1 – cos Î¸) = -cot Î¸/2 [x – a (Î¸ + sin Î¸)]

tanÎ¸/2â€‹[y âˆ’ a(2sin2Î¸/2â€‹)] = âˆ’x + aÎ¸ + asinÎ¸

tanÎ¸/2â€‹[y âˆ’ a{2(1 âˆ’ cos2Î¸/2â€‹)}] = âˆ’x + aÎ¸ + asinÎ¸

tan Î¸/2 (y – 2a) + a (2sin Î¸/2 cosÎ¸/2 = -x + aÎ¸ + a sinÎ¸

tan Î¸/2 (y – 2a) + a sin Î¸ = -x + aÎ¸ + a sin Î¸

tan Î¸/2 (y – 2a) = – x + aÎ¸

tan Î¸/2 (y – 2a) + x – Î¸ = 0

### (vi) x = 3 cos Î¸ âˆ’ cos3 Î¸, y = 3 sin Î¸ âˆ’ sin3 Î¸

Solution:

We have,

x = 3 cos Î¸ âˆ’ cos3 Î¸, y = 3 sin Î¸ âˆ’ sin3 Î¸

dx/dÎ¸ = -3sin Î¸ + 3 cos2Î¸ sin Î¸ and dy/dÎ¸ = 3 cos Î¸ – 3 sin2Î¸ cos Î¸

=  cos3 Î¸/ -sin3 Î¸

= tan3 Î¸

So the equation of the tangent at Î¸ is,

y – 3 sin Î¸ + sin3 Î¸ = -tan3 Î¸ (x – 3 cos Î¸ + cos3 Î¸)

4 (y cos3 Î¸ – x sin3 Î¸) = 3 sin 4Î¸

So the equation of normal at Î¸ is,

y – 3 sin Î¸ + sin3 Î¸= (1/tan3 Î¸) (x – 3 cos Î¸ + cos3 Î¸)

sin3 Î¸ – x cos3 Î¸ = 3sin4 Î¸ – sin6 Î¸ – 3cos4 Î¸ + cos6 Î¸

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

Solution:

Given that abscissa = 2. i.e., x = 2

x2 + 2y2 âˆ’ 4x âˆ’ 6y + 8 = 0  . . . . (1)

On differentiating both sides w.r.t. x, we get

2x + 4y dy/dx – 4 – 6 dy/dx = 0

dy/dx(4y – 6) = 4 – 2x

When x = 2, we get

4 + 2y2 – 8 – 6y + 8 = 0

2y2 â€“ 6y + 4 = 0

y2 â€“ 3y + 2 = 0

y = 2 or y = 1

m (tangent) at x = 2 is 0

Normal is perpendicular to tangent so, m1m2 = â€“1

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

The equation of normal is given by y â€“ y1 = m (normal) (x â€“ x1)

x = 2

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