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Page 1

TANGENT & NORMAL
THINGS TO REMEMBER :
I The value of the derivative at P (x1 , y1) gives the

slope of the tangent to the curve at P. Symbolically

f (x1) =
11yx

xd
yd

= Slope of tangent at

P (x1 y1) = m (say).
II Equation of tangent at (x1, y1) is ;

y y1 =
dy
dx

x y

1 1

(x  x1).

III Equation of normal at (x1, y1) is ;

y y1 = 
1

1 1

dy
dx x y

(x  x1).

NOTE :
1. The point P (x1 ,y1) will satisfy the equation of the curve & the equation of tangent & normal line.
2. If the tangent at any point P on the curve is parallel to the axis of x then dy/dx = 0 at the point P.
3. If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = or dx/dy = 0.
4. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx = ± 1.
5. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = – 1.
6. Tangent to a curve at the point P (x1, y1) can be drawn even through dy/dx at P does not exist.

e.g. x = 0 is a tangent to y = x2/3 at (0, 0).
7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the

tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation.
e.g. If the equation of a curve be x2 – y2 + x3 + 3 x2 y y3 = 0, the tangents at the origin are given by
x2 – y2 = 0 i.e. x + y = 0 and x  y = 0.

IV Angle of intersection between two curves is defined as the angle between the 2 tangents drawn to the
2 curves at their point of intersection. If the angle between two curves is 90° every where then they are
called ORTHOGONAL curves.

V (a) Length of the tangent (PT) =
 

)x(f
)x(f1y

1

2
11


(b) Length of Subtangent (MT) =

)x(f
y

1

1

(c) Length of Normal (PN) =  211 )x(f1y  (d) Length of Subnormal (MN) = y1 f ' (x1)
VI DIFFERENTIALS :

The differential of a function is equal to its derivative multiplied by the differential of the independent
variable. Thus if, y = tan x then dy = sec2 x dx.
In general dy = f  (x) d x.
Note that : d (c) = 0 where 'c' is a constant.
d (u + v  w) = du + dv  dw d (u v) = u d v + v d u

Note :
1. For the independent variable 'x' , incrementx and differential dx are equal but this is not the case with

the dependent variable 'y' i.e. y d y.

2. The relation dy = f  (x) dx can be written as
dx
dy

= f  (x) ; thus the quotient of the differentials of 'y' and

'x' is equal to the derivative of 'y' w.r.t. 'x'.

Page 2

EXERCISE–I
Q.1 Find the equations of the tangents drawn to the curve y2 – 2x3 – 4y + 8 = 0 from the point (1, 2).

Q.2 Find the point of intersection of the tangents drawn to the curve x2y = 1 – y at the points where it is
intersected by the curve xy = 1 – y.

Q.3 Find all the lines that pass through the point (1, 1) and are tangent to the curve represented parametrically
as x = 2t – t2 and y = t + t2.

Q.4 In the curve xa yb = Ka+b , prove that the portion of the tangent intercepted between the coordinate axes
is divided at its point of contact into segments which are in a constant ratio. (All the constants being
positive).

Q.5 A straight line is drawn through the origin and parallel to the tangent to a curve

a
yax 22 

= ln 

 
y

yaa 22
at an arbitary point M. Show that the locus of the point P of

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

Q.6 Prove that the segment of the tangent to the curve y =
2
a

ln
22

22

xaa

xaa




– 22 xa  contained between

the y-axis & the point of tangency has a constant length.

Q.7 A function is defined parametrically by the equations

f(t) = x =

2

1
0

0 0

2t t
t
if t

if t

 

sin
and g(t) = y =

1
0

0

2

t
t if t

o if t

sin 

Find the equation of the tangent and normal at the point for t = 0 if exist.
Q.8 Find all the tangents to the curve y = cos (x + y),  2 x  2, that are parallel to the line x + 2y = 0.

Q.9 (a) Find the value of n so that the subnormal at any point on the curve xyn = an + 1 may be constant.
(b) Show that in the curve y = a. ln (x²a²), sum of the length of tangent & subtangent varies as the

product of the coordinates of the point of contact.

Q.10 Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos2t ; y =  a cos3t contained
between the co-ordinate axes is equal to 2a.

Q.11 Show that the normals to the curve x = a (cos t + t sin t) ; y = a (sin t  t cos t) are tangent lines to the
circle x2 + y2 = a2.

Q.12 The chord of the parabola y =  a2x2 + 5ax  4 touches the curve y =
x1

1

at the point x = 2 and is

bisected by that point. Find 'a'.

Q.13 If the tangent at the point (x1, y1) to the curve x
3 + y3 = a3 (a  0) meets the curve again in (x2, y2) then

show that
1

2

1

2
y
y

x
x

 = 1.

Q.14 Determine a differentiable function y = f (x) which satisfies f ' (x) = [f(x)]2 and f (0) = –
2
1

. Find also the

equation of the tangent at the point where the curve crosses the y-axis.

Page 19

EXERCISE–III
Q.1 A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area

is to be removed from the sheet so that the vessel has maximum volume. [ REE '97, 6 ]

Q.2(a) The number of values of x where the function f(x) = cos x + cos  2 x attains its maximum is :
(A) 0 (B) 1 (C) 2 (D) infinite

(b) Suppose f(x) is a function satisfying the following conditions :

(i) f(0) = 2, f(1) = 1 (ii) f has a minimum value at x =
5
2

and

(iii) for all x f  (x) =
bax21b2ax2)bax(2

11bb
1bax21ax2ax2






Where a, b are some constants. Determine the constants a, b & the function f(x).
[JEE '98, 2 + 8]

Q.3 Find the points on the curve ax2 + 2bxy + ay2 = c ; c > b > a > 0, whose distance from the origin is
minimum. [ REE '98, 6]

Q.4 The function f(x) =

1

x

t (et  1) (t  1) (t  2)3 (t  3)5 dt has a local minimum at x =

(A) 0 (B) 1 (C) 2 (D) 3
[ JEE '99 (Screening), 3]

Q.5 Find the co-ordinates of all the points P on the ellipse (x2/a2) + (y2/b2) = 1 for which the area of the
triangle PON is maximum, where O denotes the origin and N the foot of the perpendicular from O to the
tangent at P. [JEE '99, 10 out of 200]

Q.6 Find the normals to the ellipse (x2/9) + (y2/4) = 1 which are farthest from its centre. [REE '99, 6]

Q.7 Find the point on the straight line, y = 2x + 11 which is nearest to the circle,
16 (x2 + y2) + 32 x  8 y  50 = 0. [REE 2000 Mains, 3 out of 100]

Q.8 Let f (x) = [ | | | |x for xfor x
0 2

1 0
 

. Then at x = 0, ' f ' has :

(A) a local maximum (B) no local maximum
(C) a local minimum (D) no extremum.

[ JEE 2000 Screening, 1 out of 35 ]
Q.9 Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle

of sides 'a' and 'b', the right angles of the triangle coinciding with one of the angles of the rectangle.
[ REE 2001 Mains, 5 out of 100 ]

Q.10(a) Let f(x) = (1 + b2)x2 + 2bx + 1 and let m(b) be the minimum value of f(x). As b varies, the range
of m (b) is

(A) [0, 1] (B) 0
1
2
,



(C)
1
2
1,





(D) (0, 1]

(b) The maximum value of (cos1) · (cos2).......... (cosn), under the restrictions

O < 1, 2,..............., n <

2

and cot 1 · cot 2.......... cot n = 1 is

(A)
1
2 2n/

(B)
1
2n

(C)
1
2n

(D) 1

[ JEE 2001 Screening, 1 + 1 out of 35 ]

Page 20

Q.11(a) If a1 , a2 ,....... , an are positive real numbers whose product is a fixed number e, the minimum value of
a1 + a2 + a3 +....... + an–1 + 2an is
(A) n(2e)1/n (B) (n+1)e1/n (C) 2ne1/n (D) (n+1)(2e)1/n

[ JEE 2002 Screening]
(b) A straight line L with negative slope passes through the point (8,2) and cuts the positive coordinates axes

at points P and Q. Find the absolute minimum value of OP + OQ, as L varies, where O is the origin.
[ JEE 2002 Mains, 5 out of 60]

Q.12(a) Find a point on the curve x2 + 2y2 = 6 whose distance from the line x + y = 7, is minimum.
[JEE-03, Mains-2 out of 60]

(b) For a circle x2 + y2 = r2, find the value of ‘r’ for which the area enclosed by the tangents drawn from the
point P(6, 8) to the circle and the chord of contact is maximum. [JEE-03, Mains-2 out of 60]

Q.13(a) Let f (x) = x3 + bx2 + cx + d, 0 < b2 < c. Then f
(A) is bounded (B) has a local maxima
(C) has a local minima (D) is strictly increasing [JEE 2004 (Scr.)]

(b) Prove that


)1x(·x3

x2xsin  x 


 

2
,0 . (Justify the inequality, if any used).

[JEE 2004, 4 out of 60]
Q.14 If P(x) be a polynomial of degree 3 satisfying P(–1) = 10, P(1) = – 6 and P(x) has maximum at x = – 1

and P'(x) has minima at x = 1. Find the distance between the local maximum and local minimum of the
curve. [JEE 2005 (Mains), 4 out of 60]

Q.15(a) If f (x) is cubic polynomial which has local maximum at x = – 1. If f (2) = 18, f (1) = – 1 and
f '(x) has local maxima at x = 0, then
(A) the distance between (–1, 2) and (a, f (a)), where x = a is the point of local minima is 52 .

(B) f (x) is increasing for x [1, 52 ]
(C) f (x) has local minima at x = 1
(D) the value of f(0) = 5

(b) f (x) =





3x2ex
2x1e2
1x0e

1x

x

and g (x) =  
x

0

dttf , x  [1, 3] then g(x) has

(A) local maxima at x = 1 + ln 2 and local minima at x = e
(B) local maxima at x = 1 and local minima at x = 2
(C) no local maxima
(D) no local minima [JEE 2006, 5marks each]

(c) If f (x) is twice differentiable function such that f (a) = 0, f (b) = 2, f (c) = – 1, f (d) = 2, f (e) = 0,

where a < b < c < d < e, then find the minimum number of zeros of   )x("f).x(f)x('f)x(g 2  in the
interval [a, e]. [JEE 2006, 6]

Page 37

Q.123 For the function f(x) = x4 (12 ln x  7)
(A) the point (1,7) is the point of inflection (B) x = e1/3 is the point of minima
(C) the graph is concave downwards in (0, 1) (D) the graph is concave upwards in (1,)

Q.124 The parabola y = x2 + px + q cuts the straight line y = 2x  3 at a point with abscissa 1. If the
distance between the vertex of the parabola and the xaxis is least then :

(A) p = 0 & q =  2
(B) p =  2 & q = 0
(C) least distance between the parabola and x axis is 2
(D) least distance between the parabola and x axis is 1

Q.125 The co-ordinates of the point(s) on the graph of the function, f(x) =
x x3 2

3
5
2

 + 7x - 4 where the

tangent drawn cut off intercepts from the co-ordinate axes which are equal in magnitude but opposite in
sign, is

(A) (2, 8/3) (B) (3, 7/2) (C) (1, 5/6) (D) none

Q.126 On which of the following intervals, the function x100 + sin x 1 is strictly increasing.
(A) ( 1, 1) (B) (0, 1) (C) (/2, ) (D) (0, /2)

Q.127 Let f(x) = 8x3 – 6x2 – 2x + 1, then
(A) f(x) = 0 has no root in (0,1) (B) f(x) = 0 has at least one root in (0,1)
(C) f (c) vanishes for some )1,0(c (D) none

Q.128 Equation of a tangent to the curve y cot x = y3 tan x at the point where the abscissa is

4

is :

(A) 4x + 2y =  + 2 (B) 4x  2y =  + 2 (C) x = 0 (D) y = 0

Q.129 Let h (x) = f (x)  {f (x)}2 + {f (x)}3 for every real number ' x ' , then :
(A) 'h ' is increasing whenever ' f ' is increasing
(B) 'h ' is increasing whenever ' f ' is decreasing
(C) ' h ' is decreasing whenever ' f ' is decreasing
(D) nothing can be said in general.

Q.130 If the side of a triangle vary slightly in such a way that its circum radius remains constant, then,
da
A

db
B

d c
Ccos cos cos

  is equal to:

(A) 6 R (B) 2R (C) 0 (D) 2R(dA + dB + dC)
Q.131 In which of the following graphs x = c is the point of inflection .

(A) (B) (C) (D)

Q.132 An extremum value of y =
0

x

 (t  1) (t  2) dt is :

(A) 5/6 (B) 2/3 (C) 1 (D) 2

Page 38

Q.1BQ.2DQ.3BQ.4BQ.5DQ.6CQ.7D
Q.8AQ.9DQ.10CQ.11BQ.12AQ.13DQ.14D
Q.15CQ.16DQ.17AQ.18DQ.19AQ.20AQ.21C
Q.22AQ.23CQ.24DQ.25DQ.26AQ.27BQ.28C
Q.29BQ.30CQ.31CQ.32CQ.33DQ.34DQ.35D
Q.36BQ.37AQ.38AQ.39DQ.40BQ.41AQ.42C
Q.43CQ.44CQ.45BQ.46CQ.47CQ.48BQ.49B
Q.50CQ.51AQ.52BQ.53CQ.54CQ.55CQ.56D
Q.57DQ.58CQ.59CQ.60CQ.61BQ.62AQ.63D
Q.64CQ.65AQ.66CQ.67DQ.68BQ.69DQ.70C
Q.71BQ.72AQ.73AQ.74BQ.75BQ.76AQ.77D
Q.78DQ.79AQ.80AQ.81DQ.82BQ.83AQ.84A
Q.85BQ.86AQ.87DQ.88CQ.89DQ.90CQ.91A
Q.92AQ.93BQ.94DQ.95CQ.96CQ.97CQ.98B
Q.99DQ.100DQ.101BQ.102DQ.103CQ.104D
Q.105D
Q.106BQ.107CQ.108DQ.109BQ.110BQ.111C
Q.112D
Q.113A,BQ.114A,BQ.115A,DQ.116A,CQ.117A,B,C,DQ.118
B,C

Q.119A,BQ.120A,B,CQ.121A,BQ.122A,C,DQ.123
A,B,C,DQ.124B,D

Q.125A,BQ.126B,C,DQ.127B,CQ.128A,B,DQ.129A,C
Q.130C,D

Q.131A,B,DQ.132A,B