Distance between 2 Points : Examples Video Lecture | Mathematics (Maths) Class 11 - Commerce

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before watching this video please make
sure that you have watched part 1 and
part 2
let's take home example we have to find
whether the points a B and C are
vertices of a right triangle here also
will do same thing we will first find
ABC the distance ay B BC and CA and then
we will check whether this ABC the
Pythagoras theorem is applicable if yes
we'll say that they are but is a right
triangle if no that means they are not
the vertices of a right triangle so
let's find a B Square first a and B so
let's say a B square is nothing but 10
minus 3 Square this minus this plus 20
minus 6 square plus 30 minus 9 square
this came out to be 10 minus 3 is 7
square plus 20 minus 6 14 square plus 30
minus 9 21 square and this came out to
be 49 plus 196 plus 4 4 1 and that value
is 6 Q 6
now let's find B XI Square
so if you see BC square between B and C
that is nothing but 25 minus 10 square
plus -41 minus 20 square plus 5 minus 30
square it came out to be 25 minus 10
that is 15 square my plus plus minus 60
1 square is 60 1 square plus 25 square
and that value came out to be that value
is 15 square is 225 61 square is 3 7 to
1 plus 25 square is
6:25 that value is 4 5 7 1 similarly
they'll find CA square between a and C
that comes out to be 25 minus 3 square
plus minus 4 T 1 minus 6 squared plus 5
minus 9 square correct that value is
nothing but 25 minus 3 is 22 square plus
this is 47 square plus 4 square plus 5
minus x minus 4 minus 4 square city but
for square so this came out to be 484
Plus this value is 2 to 0 947 Square and
this is 16 whose value is to 7:09
correct now let us take the biggest one
this is the biggest one so if you can
prove that VC square is equal to C a
square plus a B squared that means we
can prove it's a right triangle now
let's let's see what is C a square plus
B Square C square plus EB square the
value is 6 86 plus 2 7:09 and that is
nothing but this is not equal to 4 7 5
because we told that BC square go to 4 7
5 1 so this equation is not valid so it
is not part or right triangle and that
is our answer what we have done we have
found a B Square B Square C square and
we tried to rule to see if C the biggest
side that came out was BC square then
Vichy squares were to see a square plus
EB square or not we found that it was
not so we have proved that this is not a
right angle let's take for example you
have to find the equation of set of
points P such that P Square plus P B
Square this sorry P Square plus PB
square is equal to 2 K square this is
the portion where a B are the points
this
so is three four five and B is minus one
three and seven
so if you see PA let P be XY VV x y z eb
XY and this is the point of peace so if
you say PA you want to find PA square is
nothing but X minus 3 square plus y
minus 4 square plus Z minus 5 square
correct similarly P V Square will come
out to be X minus -1 is X plus 1 square
plus y minus 3 square plus Z minus minus
7 it is Z plus 7 is y so we have got PA
square and we got DV square now we had
told that PA square plus PB square value
to ki square so we'll add these two so
what we get is X minus 3 square plus X
plus 1 square plus y minus 4 square plus
y -3 square plus Z minus 5 square plus Z
plus 7 square is equal to 2 K square
this is the equation actually if you
want we can solve this further so this
becomes X square minus 6x plus 9 this
becomes X square minus 2x plus 2x plus 1
this becomes Y square plus 16 minus 8x
this becomes plus y square plus 3 square
is 9 minus 2 into 3 6x plus this becomes
Z square plus 25 minus 10 Z and again
this becomes plus Z square plus 49 plus
14 said this is equal to 2k so if we see
X square X square that is 2 X square
will take this value X square X squares
2x
when 2 X square plus similarly y square
Y squares 2i square 2 y square Z squared
Z square is 3 to that square plus 2 Z
square now I will take the expert 6x
minus 6x plus 2x minus 4 X minus 4 X so
this and this is minus 4 X this is y
actually minus 8y in minus 16 by that is
minus 14y this comes minus 14y this part
is taken here and z part minus 10 then
plus 14 Z is equal to plus holds it and
this is nothing but 2 k square- will add
all this value 9 + 1 10 10 and 10 this
is 9 1919 and 25 for t4 and this is 49
so this becomes total I think 1 not and
that is done that's correct
what you have done here I have just
found the value of P s square + PB
square and I said PA square was PB
squares will do case where and we got
the equation let's take for example we
have to prove that these are the
vertices of a parallel to prove this
let's suppose this is point a B C and B
correct you have to find the value of a
B BC CD b b b c CD and DA so if you can
prove that a b is equal to CD and vc is
equal to da then we can say that this is
a so first let's find the value of a B
so a B is nothing but root of 1 plus 1
square 1 plus 1 square plus minus 2
minus 2 square plus 5 minus 1 spa that
is root of 4
- 4 plus 4 is 16 plus 4 square is 16
that is 16 32 plus 36 and that is 6
similarly let's find BC BC is this and
this so this becomes root of 4 minus 1
square plus minus 7 plus 2 square plus 8
minus 5 square that came out to be to
Talk 4 minus 1 square is 9 3 squares
were nine minus five seven plus 2 is 5 5
minus 5 square is 25 and 8 minus 5 is 3
3 square is 9 so this is nothing but
root of 45 that is about 45 now they'll
find CD CD is nothing but root of 2
minus 4 square plus -3 plus 7 square
plus 4 minus 3 square that value came
out to be 2 minus 4 is minus 2 square is
4 minus 3 plus 7 is equal to 4 for
square 16 4 minus 8 is minus 4 minus 4
square of 16 that is a root of 36
similarly let me find da the value of D
just find the value of da da is nothing
but root of -1 minus u square plus 2
Plus 3 2 minus minus 3 is plus 3 square
plus 1 minus 4
correct so this came out to be root of
minus 1 minus 2 the minus 3 square that
is 9 2 plus 3 is 5 5 square is 25 and -1
minus 4 is minus 3 squares right so this
is nothing but root of again forty nine
nine nine nine 18 plus 25 root of 43
this is also 43 actually so if you see
here we can say that a B is equal to CD
is equal to six and we see is equal to
da is equal to root of forty thus we see
that a B is equal to CD is equal to six
and BC is going to be able to important
it thus it is not like this is a
parallelogram because if you see that
this is a B C D here we are seeing a B
is equal to C D and D C is equal to BC
is going to be a so this is nothing but
a parallelogram since we have proved
that these values are what is tougher
parallelogram in this question you have
to find the equation of the set of
points which are equidistant prom this
is the suppose a and this is B from the
iteration from a and B so what we can do
we can take this point let's suppose P
is equal to XYZ and then you can find PA
and PB and we can equate so you see PA
is nothing but X minus 1 square plus y
minus 2 square plus Z minus 3 square
this PA square so many PB square if you
see that is nothing but X minus 3 X
minus 3 square plus y minus 2 square
plus Z minus minus 1 square correct now
since we are told that P is equal to PB
we can also say that PA square is equal
to PB square u square both side correct
so what we get is X minus 1 square plus
y minus 2 square plus Z minus 3 square
is equal to X minus 3 square plus y
minus 2 square plus Z plus 1 square by
minus 2 square minus 2 squares cancel
and now we can solve this so this
becomes X square plus 1 minus 2 X plus Z
Square this Zed was I think Z minus 3
correct yes so this becomes Z square
plus 9 minus 6
Z is equal to dou X square plus 9 minus
6x plus Z squared plus 1 plus 2 Z so you
see X square Z square x squared square
cancel plus 9 plus 9 cancel plus 1 plus
1 cancel so what we get is this X will
come here 6x minus X is equal to 2 X
equal to 4x okay 4x is equal to this
will go there so this becomes 6 X 6 Z
plus 2 Z let's go to AIDS in or we get X
this is the equation very simple what we
have done we have taken a point P X Y Z
we found pH square we found PB square
and we told PS 4 L will do PB square and
got the answer
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