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Three dimensional geometry
Mathematics class 11 (041)
University of Kerala
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THREE DIMENSIONAL
GEOMETRY
In NDA exam, generally 3-6 questions are asked from this chapter which are based on finding coordinates of a point , direction ratios/cosine in different condition, equation of a line lplane/sphere, angle between planes/lines, radius/centre of sphere, image of a point, ete.
To represent a point in a, space we use a, y and z-coordinates, it is known as three dimensional geometry. In 3-D, there are three mutually perpendicular straight lines XOX', YOY' and 20Z' called axes Coordinates of a Point in a Space Three mutually perpendicular lines in space divide the space into 8 octants. Let the lines be XOX', YOY' and ZOZ' intersecting at O. XOY is called the XY-plane, YOZ is called the YZ-plane, ZOX is called the ZX-plane and all three planes taken together are called the coordinate planes. If P(x, y, z) is a point in space. Then,
P (x. y. z2) X-- X x= distance from YZ-plane, yl= distance from ZX-plane, and z= distance from XY-plane. Distance Formulaa (i) The distance between two points P(x1, J1, z )and Q(x2, y2, Z2) is given by PQ= V(x2 -x,+(y2 -n +(72 -2, (ii) Three points P(x1, 1, Z1 ), Q(x2, y2, Z2 ) and R(x3» y3» Zy ) are collinear, if PR = PQ+QR Note Let O be the origin and Pa. y. 2) be any point, then OP -ya y+?
Section Formulaee Let P(x, Yy z) and Q(x2, y2» Z2) be two points in space and let R be a point on the line joining P and Q such that segment C
219
MATHEMATICS
Three
Dimensional
Geometry
) It divides
/PQ internally
in the ratio m: n. Then,
the
coordinate
of R are
mx
+nx
my
- ny
mz
- nz1|
X 3 1
m+r
m+n
m+n
Direction
Cosines
(i) PQ externally
in the ratio m: n(m*
n). Then,
the
coordinates
of R are
lf a line
makes
angles
a, B, y with
the positive
directions
of X-axis,
Y-axis
and Z-axis
respectively,
then cos a,
cos B, cos y are called
its direction
cosines.
The direction
cosines
are generally
denoted
as 1, m n i.
l= cos o, m=cos
ß, n= cos Y
m*2-x
my -ny
m22n
m- n
m-n
m-n
Note
Mid-point
of PQ = t
*z, Y t y2, 2 t
Z
2
2
The angle o, ß, y are known
as direction
angles.
If the ratio in which a point
divides
a line comes
out to be
positive,
then it divides
the line internally
and if it comes
out to negative,
then it divides
the line externally
If A (xY1).
B (x2. y2. Z2) and C
(x3. Y3. 2g) represents the
vertices
of a triangle,
then the centroid
of triangle
is given by
Direction
Ratios
Three
numbers
a, b, proportional c
to the direction
cosines
, m, n of a line
are known
as the direction
ratios
of the line. Thus,
a, b, c are the direction
ratios
of
3
a line
provided
=
_
EXAMPLE
- Let Pla, b, c), Qla +2, b+2,c-2)
and Ra +6, b +6,
c -6)
be collinear
Consider
the following
statements
. R divides
PQ internally
in the ratio 3 2.
II. R divides
PQ externally
in the ratio 3: 2.
II1. Q divides
PR internally
in the ratio 1: 2.
SOME
IMPORTANT
RESULTS
(i) If OP is a directed
line segment
with
direction
cosines,
m, n such that
OP = r. Then,
the coordinates
of P
are (r, mr, nr).
P
Which
of the above
statement(s)
is/are correct?
(i) If 1, m, n are the direction
cosines
of
then + a directed line segment,
m+ n =
or cos a + cos B + cos y =
(i) DC's of Xaxis
are 1, 0, 0.
DC's of axis are 0,
1, 0.
DC's of Zaxis
are 0, 0,
a. Only
b. Only II C. land ll d. ll and Il
Sol. d. Since,
Pla, b, c), Qla
2, + b + 2, c- 2) and
Rla +6, b+6,
C-6) are collinear.
a+ 2 -Ad-
a+
)
Also,
3-
/C% y)
34b+2)-
20 b+ 6
b
(iv)I=
ya+b+
m
m=-
3-
3c-2) 2
c-
B( Y2)
and
(v) Direction ratios
of a line joining
the points
A1, 1, Z1)
and Bx2, y2» 32) are x
- X1, J
- 1 and z
- Z and
3-
So, R divides
PQ
externally
in the ratio 3
AX Y1)
its direction
cosines
are
X2-
2-
2-
AB
AB|
AB
(vi) If Plxj.)z)
and Qlx2)
are two points,
such that
the direction cosines
of PQ are l,m,n.
The,
(a +6)+
2a
Also,-
3
= a+ 2
(b+6)
- 2 -b+ 2 and
(c-6)
- 2 C - 2
Thus, Q divides
PR internally
in the ratio 1:2.
Hence,
Statements
ll and
ll are correct.
These
are projections
of PQ on X,Y and Z-axes,
respectively.
Area
of Triangle
EXAMPLE
- The direction
cosines
of a line equally
inclined
to all the three
rectangular
coordinate
axes are
Let A(x,
z1)
B(xz2,
y Z2) and Clxy.
3» Z3) be the
vertices
of a triangle,
then A = yA
+A, +A, when
.z
Z 1
b. 1,1,
d. None of these
Sol.
a. l=m
=n and + m' +n = 1
####### 3P-1l=t
####### n=+
MATHEMATICS
Three
Dimensional
Geometry
221
A PLANE plane is a surface such that any
two points lying
on
it when
joined by a line lies completely
on it.
EXAMPLE
- What
is the equation
of the plane
passing
through
(x, y1, Z,) and normal
to the line with
<a, b, c > as direction
ratios?
a. ax+ by
- cz= ax +by,
- CZ
b. alx+ ) +b(y
- y})
- c(z
- z) = 0
Equations
of Plane
in Different
Form
C. ax+ by
- CZ = 0
d. ax+by
+CZ=
+y1t Z
- The general
equation
of the first degree
in x, y,z,
i. ax + by
- cz
- d =0, represents
a plane
in which
a, b, c are constants,
and a +b
- c 0, i.
Sol. a. The equation
of the plane passing
through
(1, Y1, Z1) and
normal
to the line with
a, < b, c> as direction
ratios, is
ax
- )
- by
- y)
- dz
z1) = 0
ax-a+
by
- bY
- CZ -cZ = 0
4, b, c # 0.
- The equation
of the plane
passing
through
the
origin
is given
by ax + by
- cz = 0.
ax+ by + cz = aq
- byi
- CZ
which
is required
equation
of plane.
- Equation
of a plane
through
point (x1, y1,
Z )is
Angle
between
Two
Planees
a (x-x)
- b(y- y1)
- c(z
- z,)=
- If , m n
are the direction
cosines
of the normal
to
the plane and p
is the perpendicular distance
of the
plane
from
origin,
then the equation
of the plane
ABC will be
Let the two planes
be a^x + by+C1z
+d42X+
b2y +C2Z
- d = 0
where
(a, b, C) and (42, b2, c2) are the DR's
of normal 1) .G)
to and the planes
(i) and (ü), respectively.
Let 0 be the angle
between
the planes
(i) and (i), then
a
bb,
GC
lx + my + nz = p
cost-
- Let the plane
ABC
cut the axes OX, OY, OZat
A, B, Crespectively.
a +b +ci yaj
- bj + c
If given
planes
are perpendicular
to each other,
then
90°
cos6 = 0
4142
- bb +
It the given
planes
are parallel,
then their normals
are also
parallel,
i. the directon
cosines
of the normal
are
Let OA = a, OB = b, OC = c. Then,*
1
.e.
C(0,0,c)
proportional
Ala.
Distance 1.
of a Plane
from
a Point
Distance
or perpendicular
distance
of plane a
ax + by +
cz +d = 0 from a
point
Px 1 Z1) is given
by
P- ax +by+ +atd
a +b +c
YB(0,b)
is the required
intercept
form of the plane.
- Equation
of XY-plane
is z = 0
- Equation
of YZ-plane
is x = 0.
- Equation
of XZ-plane
is y =0.
- Equation
of plane
parallel
to YZ-plane
and at a
Distance
between
Two
Parallel
Planes
distance
a is x = a.
- Equation
of plane
parallel
to ZX-plane
and at a
distance
bis y = b.
- Equation
of plane
parallel
to XY-plane
and at a
distance
c is z = C.
Distance
between
two parallel
planes
ax +by +
cz +d1,=
and ax + by
- cz
d2 + = is given
by Na+b+ d-d
Intersection
of Two
Planes
- Equation
of plane
passing
through
the three
points
A(x1,
y1,Z
) B(x2,
)2,22)
and C{x3, 3, Z3)
If ax +b, y + c^z d, + = 0 and a^x + b2y
C2Z + +d =
x-X
Z-
x-X2 y-Y
7-7)
x-X
-ys
Z- Zs
represents
two different
planes,
then equation
of plane
passing
through
the intersection
of these
planes
is
given by (ax
b + y+ cz +d,)+a
(a,x
ba + y + C2Z +d2)
= 0
####### 222 NDA/NA Pathfinder
Angle between a Line and a Plane 2- 2-1 Z2 - Let the equations of the a line and a plane be = 1-441 and ax + by + cz +d =
m m d m2 n respectively, then the angle 0 between them is given by al + bm+ cn m 7 n
sin = a +b+c 2+m +n? m If a line is parallel to the plane, then = 0 al + bm+ cn = 0 and if line is perpendicular to the plane, then 7
SPHERE A sphere is the locus of a point which moves in space in such a way that its distance trom a tixed point always
= n EXAMPLE 7. The angle between the line 6x = 4y 3z remains constant. and the plane2-is (G) The general equation x++z2 + 2Hx+ 2v + 2wz +d = b. 0° C. tan represents a sphere with centre = (-4,-0, -w)and /2 a
a. 45° Sol. b. Angle between the plane and line is radius = vu +v +w2-d al+ bm + Cn (i1) Let P(x, y, z) be any point on the surface of a sphere whose centre is (4, b, c) and radius r, then by definition its equation 1s
Sin 6 a+b + P+m? +n? (x-a)+(y -b) + (z -c)=r If centre is origin, then the equation of reducesto sphere x++z' =r [putting a = b= c=0 in Eq. (0)
Equation of line = and equation of plane 3x+ 2y- 3z- 4 = 0 al+bm+ cn = 2x 3+ 3x 2- 4 x 3 = sin 8 =0 » 0 =0°
(1) If (x1, is Z1) and (x2, J2, Z2) be the end points of diameter of any sphere, then the equation of sphere is given by (x-x, *-2)+(y-1 My-2) t(z -z, )(z-Z2)= (iv) (x-x, )(x-x2)+(y-y10y-V2)+(z-zj) (Z-Z2)=0 represents a sphere with ends of diameter as (x1, 1 Z1) and (x2, J2, Z2) Its centre 3 22 2,
Coplanarity of Two Lines The two lines 2-23 and
-2 2 -22 are coplanar, if m2 T X2-1 y2-1 Z2-Z mm 0 m2 n EXAMPLE 8. Find the equation of the sphere having centre (-2,2, 3) and passing tn a. through the point (3, 4,- x+y2+z+ 4x+ 4y + 62 + 28 0 b. +y2 +z*-4x-4y - c. 62+ 28 x+y'+z'+ 4x+ 4y +6z -28 = 0
and the equation of plane containing them is *-X1 - 1 z-z -2 -2 z-Z m Oor m = 0 m m2 n Skew-lines The straight line which are not parallel and d. None of the above non-coplanar, i. non-intersecting are called skew-lines. Sol. d. (x+2 The equation of the sphere with centre (-2, 2, 3) 1S +(y - 2 + (z -3 = r Radius, r= 3+ 2 +(4 - 2 +(-1-3 = Required equation of the sphere is 45 (x+ 2+(y-2 + (z - 3 = (W x+y + zi + 4x -4y -6z -28 0
Shortest Distance between Two Skew-lines The shortest distance between the lines x-X1 -1-2* m and 2 -22 is given by Note Condition for orthogonal 2u4u2+2v,V2 + 2W,W2 intersection of two sphertE m2 n2 = d,+d
/
MATHEMATICS> Limits, Continuity and Differentiability 237
IMPORTANT RESULTS RELATED TO LIMIT Sol. d. We know that, - 1ssin x S 1 for all x -757sin x S7 r-7 Sx + 7 sin x [ x + Now, dividing throughout by (- 2x + 13, we get
() lim S(x) t ¢(x)]= lim flx) t lim o (x) (ii) a lim f(x)] = c lim a f(x), where c is a constant. (in) lim fx) o(x)] = lim f(x) lim o(x) 2x+3 - 2x+ 13 27+13Or all x that are large.
(iv) xa lim J«) o(x) provided *) lim ¢(x) * 0 Now, lim X- lim o (x) 23x + 13 lim -2- (v) lim logf(x) = log| lim f(x)), providedlim xa f(x)> x+7 10 (vi) limel (a) = m, -2x +13 -2+ s (x)) and lim
(vii) lim [1+ f(x)]l/* (a) = ** ( Limit of a Rational Function (vii) lim fx)] () ={ lim flx)}' lim g(x) pMx) Limit of a rational function flx), of the form lim (ix) If lim fld x)] = f [ lim x)) provided f can be find out using factorisation or substitution is continuous at o(x)¬ R methods.
2kr+3, ifx <1 Indeterminate Form EXAMPLE 1. If f (x) is defined as f()=-kx?, if a>>1 If the expression obtained after substitution of value of the limit give the following forms 0 ,0/0, 1*, - oo/ , 0 x co and o, then it is known as an indeterminate
then for what values of k does lim exist? a. b. 1 form. 1 d. -k
Sol. d. We have, lim f () lim f (1-h) = lim [2k (1- h)+ 3] EXAMPLE 3. If *>1 lim -1 tk?k lim then find h the value of k. and lim f (x) = = lim [2k (1-0)+ f(1+ h) 3]= = lim[1-k 2k +3 (1+ h) a. 0 h0 h =1- k (1+ 0) = 1-k c. 0 and 3 d Now, lim f(x) exists, if lim_ f(x) = lim, f() - Sol. b. We have, lim r1 -1 - 1 = Iim r- k 2k + 3 1- k 3k = - 2 k=- lim ( +1) - (x + 1) (r-1) lim Sandwich Theorem - (or Squeeze theorem) lim(x + 1) (x+ 1) = lim t+ k If f, g and h are functions such that f(x)S g(x)S Hx)
for all x in some neighbourhood of the point a (except possibly at x = a) and if lim f(x)=l= lim Kx) then lim fx)=l.
(1+1)(1+ 1)-
2k
3A 3 - 8 0 EXAMPLE 2. The value of lim *+7 sinx using -2x + sandwich theorem, is (3 - 8) 0 =>A =0. d-2 Since,
k =0 does not satisty the given evquation. theretore b. C k = 8/3.
NDA/NA Pathfinder 238
RILi Exponential and Logarithmic Limits and For finding the limits of exponential and logarithmic functions, following results are useful
IHIRH S, the limit dos not exist EXAMPLE 6. Evaluate lim (secx -tanx) d
d log, a, a>0 2. lim- 1 o. b. 1 C.
3. im (1+ ) Jim1-e 4. lim- e"
SO. a. lim (sec x- an x)= lim
'
- lim log0,(m>0) -sin log, 0+ x) Jog, e, (a> 0, a 1) lim (1 snX|lim lim 6, lim
- lim log+ -1 8. 1 lim (1+ x) CO5 2 sin lim C094 sin EXAMPLE 4. The value of lim log(5+ x)-log(5 )
d L'HOSPITAL'S RULE In this method, we first check, whether the form of the Sol. c. We have, lim log (5+ )-log (5-*) form function after substituting the limit isor not. If it is not of this form, then make necessary operation n lim the function otherwise we differcntiatc both numerator and denominator with respect to x. Differentiation can be done n number of times according to the problem. The above rule can be applied for other indctcrminate forms m such as
im P ,0 x, 1,0 and ctc.
boa im EXAMPLE 7. The value of lim sin 'x tan 'x x/5 -
Trigonometric Limits b. 2 (i) bm sinx = 0 d. 3 Sol. h. We have, lim sin xan 'x (i) lim cosx =1 (iv) lim X
(vi) im tan (1 x-V 'V1-'
Sin (v) m lue 1'Hoypital's Rule, " orm
EXAMPLE 5. The value of lim sin lim (14 (1 x') Iralionalisel
d. Does not exist
Sol. lim Sin IHIlim sin im 3 im
240 NDA/NA Pathfinder
Relation between Differentiability and Continuity
DIFFERENTIABILITY
Let k denotes the open interval (a, b) in R and ce k. Then, a function f:k> R is said to be differentiable at c, if and
SC+b)-J{O exists and is denoted by f"(c).
Generally, a function which is differentiable at a point necessarily continuous at that point, 1. differentiabilt. h>0 at a point > cont1nuity at that point, but converse is Let y= f (x) not necessary true. Then, the value f' (x%) EXAMPLE 10. The function f (x) = (x - 2)"3 is
= lim S(xo +ox)-J(To)8x>0 ) a. differentiable at x =
is called the right hand derivative of flx) at xo and b. not differentiable at r= 2 the value c. Cannot be determined d. None of the above f(x5)= im ox) -8x -flx,) ...(ii) Sol. b. f = (x- is called the left hand derivative of f(x) at xo Now, flx + 2) = (x+ 2-2)= ) (2)= (x- SOME IMPORTANT RESULTS = (2-2 = ) Trigonometric functions, inverse trigonometric functions, f (2+ h) - f (2) logarithmic functions, exponential functions and modulus functions are continuous in their domain.
We have, f (2) = lim h
(i) Every polynomial is continuous at every point of the real line. lim h- = h0 h (ii) The composition of differentiable functions is a differentiable function.
= h-0h2/3 lim (iv) If a function is not differentiable but it is continuous atta point, kink at it that geometrically point. implies there is a sharp corner ora which
is notdefined. Hence, thefunction is not differentiable at x = 2.
PRACTICE EXERCISE
a + 2x) - (3x) 5. lim COs 3r
- Ia lim (3a + x) - 2x a 0 is equal to r»0 sin x is equal to 2
(c)
(a) 1 2 (b) 3 (a) () 3/3 (d) 0 2 (d)
- 3x +2 is equal to 6. lim 2x- 7sequal to
- lim 2x +x - 3
U2 COS X (a) 1 (b) 2 (c)-2 (d) 0 (a) 2 (0) (c)C (d) 1 7. lim v2-cose sin (40 n) 2 1 0T4 is equal to 3. lim I(1+ x) - 1
I8 equal to (a) 16 (b) 1 16/ (a) log 2 (b) 2 tog 2 ()log2 (d) t 8in * 8. im 4. im (x- 3) equal to -cos 18 x) equal to x3|x -31 (a) 0 (b) 1 (c) -1 (d) t (a) 0 (b) 1 (c) -1 (d) does not oxist
Three dimensional geometry
Course: Mathematics class 11 (041)
University: University of Kerala
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