L11 : Normal Form - Straight line, Mathematics, Class 11 Video Lecture

hello friends this video straight line
point 11 is brought to you by exam feel
calm no more fear from exam before
watching this video please make sure
that you have a straight line part 1 -
part 10 now let's understand normal form
suppose there is a non vertical line
because you have in normal form if it is
a vertical line the slope is infinite so
for a non vertical line we have the
following it down we have the length of
perpendicular from origin to the line
that means if this is the line we have
this length this length we have also we
have the angle which normal makes little
first interaction of X this normal makes
angle Omega with the quasi direction of
X so we have two important one is we
have the length of this perpendicular
from origin to this line plus we have
the angle which this perpendicular makes
with the x axis if we have these two
values we can write the equation this
lime line this L as X cos Omega plus y
sine Omega is equal to P where P is the
length of this this line that is from
horizion if we draw perpendicular to
this line whatever length we get is the
P and we have the Omega angle angle that
is the perpendicular subset to the
x-axis so if we have those values then
the equation of the line will be X cos
Omega plus Phi in sine Omega is going to
do if you want we can prove this also
first let's find the slope of line help
also will say slope of line L let's
suppose this is M so we'll say M is
nothing but minus 1 by slope of line P
because both are perpendicular so minus
1 by slope of line please
tan Omega so this is the slope of line
again this we have got or we can also
say minus 10 Omega is nothing but minus
this term we can say sine Omega by so
this is one equation
their slope of line L is minus cos Omega
by sine Omega now since this is P so we
can say that this point because theta
and this point will be P sine theta so
this if you see the x-coordinate will be
B cos theta this P cos theta or P cos
omega comma P sine Omega this is the X&Y
coordinate so if you see for point a the
point will be P cos Omega and P sine
Omega this is a value of x and y 4.80
since this is P this is Omega this
distance is called P cos Omega and this
dotted distance which you see here is
nothing but P sine Omega based on the
normal techno battery so if you see here
this distance is P sine Omega and this
is P cos Omega so for this point a the X
distance is this one that is P cos Omega
and the Y distance is P sine Omega so we
have point is P cos Omega gamma P sine
Omega now this is very simple we have
slope and we have one point we can form
the equation of line so equation of the
line is nothing but Y minus P sine Omega
by X minus P cos Omega is equal to M and
M is nothing but Y minus cos Omega Y
sine Omega this is them all right now I
will do a cross multiplication so what
we get is y sine Omega minus P sine
square Omega is equal to minus X cos
Omega plus B cos square Omega or we can
say X cos Omega a on the left hand side
plus y sine Omega is going to be into
sine square Omega plus cos square theta
this is what we get all we know that
sine squared Omega plus cos square Omega
is equal to 1 so we can say that X cos
bring up les sine Omega is equal to 1
because we know that sine square theta
plus cos square theta is not true this
is what we know so we can say sine
square plus cos square into 1 so this
become 1 into B then this P and that is
our desired equation so we have found
this equation x cos omega plus y sin of
x value we have proved this proof of you
may not require question you should
remember that if you are given that for
a line if you draw perpendicular from
origin then distance is P and that the
particular mix of Omega angle with x
axis then the equation can be written as
X cos Omega
that's why sine Omega is you can do we
will take one example here it says that
the perpendicular distance from the
origin is 5 minute that this P is equal
to Phi and the angle machine the normal
makes positive x-direction is 30 degree
please note this angle is the angle
machine normal makes not the line so we
can say the equation of the line is X
cos Omega plus y sine Omega 's equal to
P or we can say X cos 30 degree plus y
sine 30 degree is equal to Phi cos 30
degree you know is root 3 by 2
and sine 30 degree we know is 1 by 2 we
have lunch in diplomatic is equal to 5
or if you solve this we get root 3x plus
y minus 10 is equal to 0 this is the
equation of the line very simple we were
told the value of P and we're told that
the value of the angle which P makes
x-axis positive x-axis if just use the
formula X cos Omega plus y sine Omega is
equal to P and we have got them thank
you
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