12 math 1st-merged

Ex 11.1 6:29 AM

Friday, July 24, 2020

1) Distance between the two points is
in the ratio of
PQ=
2) Coordinate which divides the line joining points

of

3) Coordinate mid point of the line joining points

4) If are the angle made by the lineAB with the positive direction of the X-axis ,Y-

axis ,Y- axis respectively . Then called the direction cosines of the

line AB and denoted by

5) Direction cosine of the Coordinate are
Direction cosine of X-axis are (1,0,0)
Direction cosine of Y-axis are (0,1,0)
Direction cosine of Z -axis are (0,0,1)

6) If direction cosine and length of a line are given the the co-ordinate of the point

7) Relation between the direction cosine of a line are called

8) Any quantiles a ,b ,c which are proportional to the direction cosine
direction ration

9) Direction ratio of the line two points
b

10) Direction cosine of the line two points

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11) Direction cosine of the a point OP=r is
12) Angle between the lines whose Direction cosines are given

Let be the direction cosines of the two given staringh lines AB and
CD respectively and be the angle between them . Let OP and OQ be two lines drawn
through O and Parallel to AB and CD respectively . I the coordinate of P and Q are
( respectively and

using distance formula
=

by using cosine law we have
=

comparing we get
13) Perpendicular condition
14) Parallel condition
15) Angle between the lines whose directions ratio are given
16) Perpendicular condition

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17) Parallel condition

Important Examples are Example 7,12,13,14

Exercise 11.1
1) Find the distance between the points
a) (2.1,3) and (4,3,-6)

solution
Given points are P(2.1,3) and Q (4,3,-6)
distance between two points is given by
PQ=
PQ=

2) Show that following points are collinear
(-2,3,5) ,(7,0,-1) and (1,2,3)
solution
Here A(-2,3,5) ,B(7,0,-1) and C(1,2,3)
distance between two points is given by
d=
AB=
BC=2
AC=

Hence given points are collinear
3) Find the mid points of the line joining
a) (1,-2,4) and (3,4,2)
we have

x= 2, y=1, z= 3
4) Find the coordinate of line joining that divides
a) (2,-4,3) and (-4,5,-6)

solution
given points are (2,-4,3) and (-4,5,-6)
given ratio internally =1:2
we have

x=

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y=

z=
5) Given three collinear points A(3,2,-4) ,B(5,4,-6) and C(9,8,-10) .find the ratio in which B

divides the AC
Solution
Let B divides AC in the ratio of
we have

5 b) Find the ratio in which the line joining the points (-2,4,7) and (3,-5,-8) is divides by X Y
plane

Solution
any point of X Y plane is given by (x, y ,0)
we have

6) Find the direction cosine of each of the lines whose direction ratios are given
a) 2,3,6

Solution
Given ratios are 2,3, 6

we have

7) Find the direction cosine of the line passing throw the points
a) A(-1,2,-3) and B(4,-1,1)

Solution
Given points are
Direction ratio of the line two points

a=5
b
b=-3

c=4

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8) in board
8b) If the line makes angle of 45 and 60 with the positive X-axis and Y-axis respectively ,
find the acute angle made by the line with the positive Z-axis
solution
the line makes angle of 45 and 60 with the positive X-axis and Y-axis

we have

9) Find the angle between the two lines whose direction ratio are given
a) 1,2,4 and -2,1,5

Solution
Let
we have

10) Show that the line joining the points (1,2,3) and (-1,-2,-3) is parallel to the line joining the
points (2,3,4) and (5,9,13)
Solution
Direction ratio of the line two points

b

direction cosines of line joining the points (1,2,3) and (-1,-2,-3) are

direction cosines of line joining the points (2,3,4) and (5,9,13) are

Two lines are parellel with direction ratio

so we ,have

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so we ,have

Hence proved
11) for what value of K is the line joining points (1,2,3) and (4, 5 ,k) parallel to the line
joining the points (-4,3,-6) and (2,9,2)

12) find the direction cosines of the line which is perpendicular to the lines with direction cosine
proportional to 1,-2,-2 and 0,2,1
Solution
Let the directions cosines of the line are
The line with direction cosine 1,-2,-2 is perpendicular to the line with direction cosines
by perpendicular condition
…1
The line with direction cosine 0,2,1 is perpendicular to the line with direction cosines
by perpendicular condition
now solving equation I and ii
Now cross multiplication method
0 2 10 2

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13) Find the direction cosines of two lines which satisfy the equation
.a) and …I
given equation are

from I

putting the value of in equation ii we get

Either
or
for solving equation I and iii
for cross multiplication method
0 2 102

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for solving equation I and iv we
0 1 201

let be the angle between the lines
we have

=
=

14 show that the lines whose direction cosines are given by the relation
a) Parallel if
b) Perpendicular if

solution
Given equation are

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from I
putting the value of form I to in equation ii we get

It is quadratic in

if the lines are parallel then

equal roots

Quadratic equation have equal roots if

comparing with we get

substituting value of A, B ,C on

Hence proved
14 b)
solution
Given equation are

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from I
putting the value of from I to in equation ii we get

It is quadratic in we get
comparing with

Product of root of the quadratic equation =

similarly substituting value of the m from I to ii we get

by perpendicular condition

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hence proved with four diagonlas of a cube then proves that

15 ) if a line which makes angle
a)
b)

solution
Let us take one corners O of the cube a as origin and the edges OA ,OB ,OC along the X-
axis , Y-axis and Z -axis respectively . Let OA=OB=OC=a .Then The co-ordinate of O,
A ,B, C, D ,E, F, G are O(0,0,0,)

Four diagonals of the cube are OE ,AF, BD, CG
direction ratio of a line joining any two point is given by

Direction ratio of OE=
Direction cosine of OE=
Direction ratio of FA=
Direction cosine of FA=
Direction ratio of BD=
Direction cosine of BD=
Direction ratio of CG =
Direction cosine of CG =
let are the direction of a line

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let are the direction of a line
Angle between the two lines is given

line and OE

is the angle between the
=

is the angle between the line

is the angle between the line and

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To prove
LHS

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Ex 11.2 7:33 AM

Saturday, January 15, 2022

A plane is surface such that the straight line joining any two points on it lies
entirely on the surface ,

1) Genera equation of plane in first degree is

2) Equation of a plane passing through a point

3) A plane passing throw intersection of two plane

4) Equation of plane in intercept form

5) Equation of plane in normal form
6) Equation of plane through 3 points

7) angle between two planes

8) Perpendicular condition to a plane
9) Parallel condition

10) length of the perpendicular from a point

11) Distance between two parallel planes

Exercise 11.

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Exercise 11.
1a) find the equation of the plane which makes intercepts 1,2,3 on X-axis , Y -
axis , Z- axis respectively
solution
Given intercepts are 1,2,3
Equation of plane intercepts form is

which is the required equation
1b) find the equation of the plane which makes equal intercepts on the axes and
passes through the point (2,1,1)
Solution
let
Equation of plan in intercept form is given by

it passes through (2,1,1)

k=4
putting value of 'K' in equation I

Which is the required equation of the plane

2) Reduce the equation l form and

determined the direction cosines of the normal .Also find the length of the

perpendicular to it from the origin

Solution

Given equation of the plane

Now

Dividing both sides by

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Which is the required
comparing with the
direction cosine of the normal are
length of the perpendicular from the origin =3

3) a) Find the equation of the plane passing through the points (1,1,1) (3,-1,2)
and (-3,5,-4)
Solution
Equation of the plane through one point is given by
Equation of plane t through one(1,1,1) is given by
Equation I passes through the point (3,-1,2) so

Equation I passes through the point (-3,5,-4 ) so

For solving equation ii and iii

-4 4 -5 -4 4

Let
Then

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Now substituting the value of A, B ,C we get

which is the required equation of plan

4 ) find the equation of the plane through the point (1.2.3) and is parallel to the the
plane
Solution
Given equation of plane is

Equation of plane parallel to
is given by

Since it passes through the point (1.2.3)

k=4
substituent the value of the k we get

Which is the required equation of the plane

Q5
Find the angle between the plane

Solution
Given equation of plane are

Comparing with
we get

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we get
Equation between the two plane is

=

Q 7) find the equation of the plane through the points (-1,3,2) and is

perpendicular to each of the two planes and

solution
Equation of the plane passing through the point (

Equation I perpendicular to +2y+2z=5

Equation ii is perpendicular to

now solving &

for cross multiplication method

32 2 3 2

Putting value of the A ,B, C

8)find the equation of the plane which passes through (-1,1,1) and (1,-1,1) and is
perpendicular to the plane

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perpendicular to the plane
Solution
Equation of plane through the point (

Equation 1 passes through the points (1,-1,1)
Plane I is perpendicular to
Solving by cross multiplication method
2 -2 0 2 -2
1 2 21 2

9) a) . Find the equation of the plane passing through the intersection of the plane

Solution
The equation passing through the intersection of

is given by

or
it is perpendicular to the

substituting the value of in equation I

10) .a) Find the equation of the plane through (1,2,3) and perpendicular to the
line joining the point (1,3,4) and (3,5,6)
Equation of plane through the point (

Direction ratio of line joining the points (1,3,4) and (3,5,6)

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=2
The normal of plane is parallel to the line perpendicular to it
by parallel condition

substituting value of A ,B ,C in equation I

11) Find the length of perpendicular to a plane
a) From the point (1,2,3) to the plane

length of the perpendicular from a point

=

b) .
c) find the length of the perpendicular from the points (1,4,5) on the plan

(2,-1,5)(0,-4,1) (0,-6,0)
Equation of plan passing through the three points is given by

length of the perpendicular from a point to a plane

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length of the perpendicular from a point to a plane

length of the perpendicular from a point to a plane

12) Find the value of k in the length of perpendicular from the points (2,3,4) on

the plane

Solution

length of the perpendicular from a point to a plane

4=

Squaring on both side

13) Find the distance between the the following parallel planes
solution
Given equations are

or
Distance between two parallel plan is given by

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Distance between two parallel plan is given by

14) A variable plane is at a constant distance 3p from the oringine and meets the
axes in the points A,B, C . Prove that the loscus of the centroid of the traingle
ABC is

Solution
The equation of the plane in intercept form is given by
perpendicular distance between ( 0,0,0 ) and plan is 3p so

squaring on both sides
The co-ordinate of A B,C are A( a,0,0 ) ,B( 0,b,0 ) C( 0,0,c)
Let

,
putting the value of a,b,c in equation I

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for locus of point replace by
we get

hence proved

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