TANGENT& NORMAL # 1

TANGENT & NORMAL DEFINITIONS

AND RESULTS

Let y = f (x) be the equation of a curve and let

P (x1 , y

1) be a point on it . Then derivative at

P (x1 , y

1) gives the slope of the tangent to the

curve at P .

Equation of tangent at (x1, y

1) is ;

y y1 =

f

(x

1) (x x

1) .

Equation of normal at (x1, y

1) is ;

f (x

1) (y y

1) + (x x

1) = 0 .

NOTE :

1. If the tangent at any point P on the curve is parallel to the axis of x then dy/dx = 0 atthe point P .

2. If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = ordx/dy = 0 .

3. If the tangent at any point on the curve is equally inclined to both the axes theredy/dx = ± 1 .

4. Length of the tangent (PT) =

y f x

f x

1 1

2

1

1

5. Length of Subtangent (MT) =

y

f x1

1

6. Length of Normal (PN) = y f x1 1

21 7. Length of Subnormal (MN) = y

1 f (x

1)

ANGLE BETWEEN TWO CURVES :

Angle of intersection between two curves is defined as the angle between the 2 tangentsdrawn to the 2 curves at their point of intersection . If the angle between two curves is90° then they are called ORTHOGONAL curves .

RATE MEASURE :d yd x

represents the rate of change in ' y

' with respect to '

x

' . For example if '

y

' is

displacement and ' t

' is time then

d yd t

represent the velocity .

EXERCISE I

1. The number of values of c such that the straight line 3x + 4y = c touches the curve

x4

2 = x + y is :

(A) 0 (B) 1 (C) 2 (D) 4

2. Equation of the normal to the curve y = x + 2 at the point of its intersection with thecurve y = tan (tan 1 x) is :

length of tangent

)

y = f(x)

P (x1, y1)

length of

normal

N xT

length of subtangent

)

Mlength

ofsubnormal

O

>

>

>

TANGENT& NORMAL # 2

(A) 2x y 1 = 0 (B) 2x y + 1 = 0 (C) 2x + y 3 = 0 (D) none

3. Two tangents to the graph of the function f(x) = 17 1 2 x intersect at right angles at a

certain point on the y axis . The equations of the tangents are :(A) x y + 4 = 0 (B) x + y 4 = 0 (C) x + y + 4 = 0 (D) x y 4 = 0

4. Equation of normal drawn to the graph of the function defined as f(x) = sin x

x

2

, x 0

and f(0) = 0 at the origin is :(A) x + y = 0 (B) x y = 0 (C) y = 0 (D) x = 0

5. The points of contact of tangents drawn from the origin to the curve y = sin x lie on thecurve given by :(A) x2y2 = x2 + y2 (B) x2y2 = x2 y2 (C) x2/y2 = x2 + y2 (D) x2/y2 = x2 y2

6. The line which is parallel to x-axis and crosses the curve y = x at an angle of 4

is

(A) y = 12

(B) x = 12

(C) y = 14

(D) y = 12

7. The tangent to the curve 3 xy2 2 x2y = 1 at (1, 1) meets the curve again at the point

(A)

165

120

, (B) 165

120

,

(C) 1

20165

,

(D)

120

161

,

8. The angle of intersection of the curve y = x2 & 6y = 7 x3 at (1, 1) is :

(A) 5

(B) 4

(C) 3

(D) 2

9. The equation of normal to the curve xa

yb

n n

= 2 (n N) at the point with abscissa

equal to 'a' can be :(A) ax + by = a2 b2 (B) ax + by = a2 + b2

(C) ax by = a2 b2 (D) bx ay = a2 b2

10. The lines y = 32

x and y = 25

x intersect the curve 3x2 + 4xy + 5y2 4 = 0 at the

points P and Q respectively . The tangents drawn to the curve at P and Q :(A) intersect each other at angle of 45º (B) are parallel to each other(C) are perpendicular to each other (D) none of these

11. The point(s) of intersection of the tangents drawn to the curve x2y = 1 y at the pointswhere it is intersected by the curve xy = 1 y is/are given by :(A) (0, 1) (B) (0, 1) (C) (1, 1) (D) none of these

12. A curve is represented by the equations, x = sec2 t and y = cot t where t is a parameter.

If the tangent at the point P on the curve where t = /4 meets the curve again at the pointQ then PQ is equal to :

(A) 5 3

2(B)

5 52

(C) 2 5

3(D)

3 52

13. For the curve represented parametrically by the equations ,x = 2 ln cot t + 1 & y = tan t + cot t

(A) tangent at t = /4 is parallel to x axis(B) normal at t = /4 is parallel to y axis(C) tangent at t = /4 is parallel to the line y = x

TANGENT& NORMAL # 3

(D) tangent and normal intersect at the point (2, 1)14. The curve y exy + x = 0 has a vertical tangent at :

(A) (1, 1) (B) (0, 1) (C) (1, 0) (D) no point

15. A curve with equation of the form y = ax4 + bx3 + cx + d has zero gradient at the point(0, 1) and also touches the x axis at the point ( 1, 0) then the values of x for whichthe curve has a negative gradient are :(A) x > 1 (B) x < 1 (C) x < 1 (D) 1 x 1

16. The curves x3 + p xy2 = 2 and 3 x2y y3 = 2 are orthogonal for :(A) p = 3 (B) p = 3 (C) no value of p (D) p = ± 3

17. At any two points of the curve represented parametrically by x = a (2 cos t cos 2t) ;y = a (2 sin t sin 2t) the tangents are parallel to the axis of x corresponding to the valuesof the parameter t differing from each other by :(A) 2/3 (B) 3/4 (C) /2 (D) /3

18. The ordinate of y = (a/2) (ex/a + e-x/a) is the geometric mean of the length of the normaland the quantity :(A) a/2 (B) a (C) e (D) none of these .

19. If the line , ax + by + c = 0 is a normal to the curve xy = 2, then :(A) a < 0 , b > 0 (B) a > 0, b < 0 (C) a > 0 , b > 0 (D) a < 0 , b < 0

20. Three normals are drawn to the parabola y2 = 4x from the point (c , 0). These normals arereal and distinct when(A) c = 0 (B) c = 1 (C) c = 2 (D) c = 3

EXERCISE II

LEVEL I

1. Show that the line xa

+ yb

= 1 touches the curve y = b e x/a at the point where it crosses

the y

axis .

2. Find the parameters a , b , c if the curve y = a x2 + b

x + c is to pass through the point

(1 , 2) and is to be tangent to the line y = x at the origin .

3. Show that the normal to any point of the curve x = a (cos t + t sin t) ,y = a (sin t t cos t) is at a constant distance from the origin .

4. Prove that the straight line , x cos + y sin = p will be a tangent to the curve

x

a

2

2 + y

b

2

2 = 1 , if p2 = a2 cos2 + b2 sin2 .

5. If the tangent at (x1 , y

1) to the curve x3 + y3 = a3 meets the curve again at (x

2 , y

2)

show that xx

2

1 +

yy

2

1 = 1 .

6. Show that the curves y = 2 sin2 x and y = cos 2 x intersects each other at x =

6

. Also

find the angle of intersection .

7. Find the angle of intersection of the following curves :

TANGENT& NORMAL # 4

(i) 2 y2 = x3 & y2 = 32 x (ii)x

a

2

2 + y

b

2

2 = 1 & x2 + y2 = a b

8. Show that the condition , that the curves x2/3 + y2/3 = c2/3 and x

a

2

2 + y

b

2

2 = 1 may

touch, if c = a + b .

9. An inverted cone has a depth of 10 cm & a base of radius 5 cm . Water is poured into itat the rate of 1.5 cm3/min . Find the rate at which level of water in the cone is rising ,when the depth of water is 4 cm .

10. The top of a ladder 13 m long is resting against a vertical wall when a ladder beginsto slide . When the foot of the ladder is 5 m from the wall , it is sliding at the rate of2 m/s . How fast then is the top sliding downwards ?

LEVEL II

11. Find the

equation

of

the

normal

to

the

curve x3 + y3 = 8 x

y at

the

point

other

than

origin where it meets the curve y2 = 4 x .

12. Find the equation of tangent to the curve whose parametric equations are ,

x =

a

t2cos . cos t ; y = a t2cos . sin t at t =

6

.

13. Does the straight line by

ax

=

2 touch the curve

22

by

ax

=

2 ? If it touches,

give the co-ordinates of the point of contact .

14. Show that the normal to the rectangular hyperbola x y =

c2 at the point

t

1 meets the

curve again at the point t2 such that t

13 t

2 =

1 .

15. If the point on y = x tan gx

u

2

2 22 cos ( > 0) , where the tangent is parallel to y = x

has an ordinate u

g

2

4 , then find the value of

.

16. Prove that all points on the curve , y2 = 4 a

axsinax at which the tangent is

parallel to the x

axis lie on the parabola y2

=

4 a

x .

17. If x1 &

y

1 be the intercepts on the axes of '

x

' and '

y

' cut off by the tangent to the curve,

nn

by

ax

= 1 , then show that

)1n/(n

1

)1n/(n

1 yb

xa

= 1 .

18. In the curve xa yb = Ka+b , prove that the portion of the tangent intercepted between thecoordinate axes is divided at its point of contact into segments which are in a constantratio . (All the constants being positive) .

19. Find the minimum value of (x1 x

2)2 + 2

912

2

2

x

x where x

1

0 2, and

x2

R+ .

20. Show that the distance from the origin of the normal at any point of the curve

TANGENT& NORMAL # 5

x = a e sin cos 2

22

& y = a e cos sin 2

22

is twice the distance of the tangent at the

point from the origin .21. Show that the curves y2

=

4 a

x &

a

y2 = 4

x3 intersect each other at an of

tan 1 (1/2)

and also if PG1 &

PG

2 be the normals to the two curves at common point of intersection

(which is not origin) meeting the axis of x in G1 & G

2 , then G

1 G

2 =

4 a.

22. In the curve x = a t2

1tanlogtcos , y = a sin t , show that the portion of the tangent

between the point of contact and the x

axis is of constant length .

23. Prove that the equation of the normal to ,

x2/3 + y2/3

=

a2/3

is y cos x sin

=

a cos 2 , where

is the angle which the

normal makes with the axis of ' x

' .

24. Find the abscissa of the point on the curve , x

y

= (c + x)2

the normal at which cuts off

numerically equal intercepts from the axes of co-ordinates .

25. Show that the condition that the curves , a x2

+

by2

=

1 & a

x2

+

b

y2

=

1 should

intersect orthogonally is that , b1

a1 =

b1

a1

.

26. A straight line is drawn through the origin and parallel to the tangent to a curve

x a y

a

2 2

= ln a a y

y

2 2

at an arbitary point M . Show that the locus of the

point P of intersection of the straight line & the straight line parallel to the x-axis &passing through the point M is x2 + y2 = a2 .

27. A curve is given by the equations x = at2 & y = at3 . A variable pair of perpendicular linesthrough the origin 'O' meet the curve at P & Q . Show that the locus of the point ofintersection of the tangents at P & Q is 4y2 = 3ax a2 .

28. For the curve x2/3 + y2/3 = a2/3 , show that | |z 2 + 3p² = a² where z = x + i y & p is thelength of the perpendicular from (0 , 0) to the tangent at (x , y) on the curve .

29. The tangent at a variable point P of the curve y = x2 x3 meets it again at Q . Show thatthe locus of the middle point of PQ is y = 1 9x + 28x2 28x3 .

30. Find all the tangents to the curve y = cos (x + y) , 2 x 2 , that are parallel to theline x + 2y = 0 .

31. A water tank has the shape of a right circular with its vertex down . Its altitude is 10 cmand the radius of the base is 15 cm . Water leaks out of the bottom at a constant rate of1 cu. cm/sec . Water is poured into the tank at a constant rate of C cu. cm/sec . ComputeC so that the water level will be rising at the rate of 4 cu cm/sec at the instant when thewater is 2 cm deep .

32. A circular metal plate expands under heating so that its radius increases by 2 % . Findthe approximate increase in the area of the plate , if the radius of the plate before heatingis 10 cm .

33. An air force plane is ascending vertically at the rate of 100 km/h . If the radius of theearth is R Km , how fast the area of the earth , visible from the plane increasing at 3 min

after it started ascending . Take visible area A = 2 2R hR h

Where h is the height of the

plane in kms above the earth .

TANGENT& NORMAL # 6

34. A man 1.5 m tall walks away from a lamp post 4.5 m high at the rate of 4 km/hr .(i) how fast is the farther end of the shadow moving on the pavement ?(ii) how fast is his shadow lengthening ?35. If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining

elements are changed slightly, show that d a

Acos +

d bBcos

= 0 .

EXERCISE III1. Find the equations of the tangents drawn to the curve y2 2 x3 4y + 8 = 0 from the

point (1 , 2) . [ REE ’90, 6 ]

2. Find the equation of the straight line which is tangent at one point & normal atanother point to the curve , y = 8 t3 1 , x = 4 t2 + 3 . [ REE ’91, 6 ]

3. 3 normals are drawn from the point (c , 0) to the curve y² = x . Show that c mustbe greater than 1/2 . One normal is always the x - axis . Find c for which the othertwo normals are perpendicular to each other . [ JEE ’91, 4 ]

4. [ REE ’92, 6 + 6 ](a) A ladder 15 m long leans against a wall 7 m high & a portion of the ladder protrudes over

the wall such that its projection along the vertical is 3 m. How fast does the bottomstart to slip away from the wall if the ladder slides down along the top edge of the wall at2 m/s ?

(b) Show that the normal to the curve 5 x5 10 x3 + x + 2 y + 6 = 0 at P (0 , 3) meets the curveagain at two points . Find the equations of the tangents to the curve at these points .

5. [ JEE ’93, 3 + 5 ](a) Find the equation of the normal to the curve y = (1 + x)y + sin1 (sin² x) at x = 0 .(b) Tangent at a point P

1 [other than (0 , 0)] on the curve y = x3 meets the curve again at P

2 .

The tangent at P2 meets the curve at P

3 & so on . Show that the abscissae of

P1, P

2, P

3, ......... P

n, form a GP . Also find the ratio

area P P P

area P P P1 2 3

2 3 4 .

6. Find the points on the curve 9 y2 = x3 where normal to the curve make equal interceptswith the axes . [ REE ’93, 6 ]

7. [ JEE ’94, 2 + 5 ](a) Let C be the curve y3 3 xy + 2 = 0 . If H is the set of points on the curve C where the

tangent is horizontal & V is the set of points on the curve C where the tangent is vertical,then H = _____ & V = _____ .

(b) The curve y = ax3 + bx2 + cx + 5 , touches the x - axis at P ( 2 , 0) & cuts the y - axis ata point Q where its gradient is 3 . Find a , b , c .

8. Find the acute angles between the curves y = x2 1 and y = x2 3 at their point ofintersection . [ REE '98, 6 ]

9. Find the equation of the straight line which is tangent at one point and normal at anotherpoint of the curve, x = 3 t2 , y = 2 t3 .

[ REE 2000 (Mains) 5 out of 100 ]

10. If the normal to the curve , y = f (x) at the point (3, 4) makes an angle 34

with the

positive x axis . Then f (3) =

TANGENT& NORMAL # 7

(A) 1 (B) 34

(C) 43

(D) 1

[ JEE 2000 (Screening) 1 ]11. [ JEE 2001 Screening , 1 + 1](a) The triangle formed by the tangent to the curve f (x) = x2 + b

x b at the point (1 , 1) and

the co-ordinate axes lies in the first quadrant . If its area is 2 , then the value of b is :

(A) 1 (B) 3 (C) 3 (D) 1

(b) The equation of the common tangent touching the circle (x 3)2 + y2 = 9 and the parabolay2 = 4

x above the x-axis is :

(A) 3 y = 3 x + 1 (B) 3 y = (x + 3)

(C) 3 y = x + 3 (D) 3 y = (3 x + 1)

12. [ JEE 2002 Screening , 3 + 3 + 3 ](a) If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line

5x 2y + 6 = 0 at a point Q on the y-axis , then the length of PQ is :

(A) 4 (B) 2 5 (C) 5 (D) 3 5

(b) The equation of the common tangent to the curves y2 = 8 x

& x y = 1

is :

(A) 3 y = 9

x + 2 (B) y = 2

x + 1 (C) 2

y = x + 8 (D) y = x + 2

(c) The position on the curve , y2 + 3

x2 = 12 y where the tangent is vertical , is/are :

TANGENT& NORMAL # 8

(A)

4

32, (B)

113

1, (C) (0 , 0) (D)

4

32,

ANSWERSHEET

EXERCISE I1. B 2. A 3. AB 4. A 5. B 6. D 7. A8. D 9. AC 10. C 11. B 12. D 13. AB 14. C15. C 16. B 17. A 18. B 19. AB 20. D

EXERCISE II

2. a = 1 , b = 1 , c = 0 6. 3

or 23

7. (i) tan 1 12

(ii) tan 1 a b

a b

9. 3/8 cm/min 10. 5/6 m/s

11. y = x 12. y =

a

42

13. yes , (a, b) 15. 3

19. 2 2

24. ± 2

c 30. x + 2 y = /2 & x + 2 y = 3 /2 31. 1 + 36 cu. cm/sec

32. 8 sq. cm 33. 200 R3 / (R + 5)² km² / h 34. (i) 6 km/h (ii) 2 km/hr

EXERCISE III

1. 2 3 x y = 2 3 1 or 2 3 x + y = 2 3 1

2. 2 x y = (89 2 /27) + 1 or 2 x + y = (89 2 /27) 1 3. c = 3/4

4. (a) 6

5 m/s (b) normal at P (0 , 3) intersects the curve at A (1 , 1) & B ( 1 , 5) .

The equation of the tangent at both A & B is 2 x y = 3

5. (a) x + y 1 = 0 (b) 1/16 6. (4, 8/3) 7. (a) H = ; V = {(1 , 1)}

(b) a = 1/2 ; b = 3/4 ; c = 3 8. = tan1 4 2

7

9. 2 x + y 2 2 = 0 or 2 x y 2 2 = 0 10. D

11. (a) C (b) C 12. (a) C (b) D (c) D