JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Examples : Tangents and Normals
Examples : Tangents and Normals Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Examples : Tangents and Normals Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is a tangent line? |  |
Ans. A tangent line is a straight line that touches a curve at a specific point, known as the point of tangency. It has the same slope as the curve at that point.
| 2. How do you find the equation of a tangent line? | |
Ans. To find the equation of a tangent line to a curve at a given point, you need to find the derivative of the curve, evaluate it at the given point to find the slope, and then use the point-slope form of a line to write the equation.
| 3. What is the difference between a tangent and a normal line? | |
Ans. A tangent line touches a curve at a specific point and has the same slope as the curve at that point, while a normal line is perpendicular to the tangent line and has a slope that is the negative reciprocal of the tangent line's slope.
| 4. How do you determine if two lines are tangent to a curve? | |
Ans. Two lines are tangent to a curve if they touch the curve at the same point and have the same slope as the curve at that point. You can determine this by finding the derivative of the curve and evaluating it at the point of contact.
| 5. Can a curve have multiple tangent lines at the same point? | |
Ans. No, a curve can have only one tangent line at a specific point. The tangent line represents the instantaneous rate of change of the curve at that point, and there can only be one unique rate of change at any given point.
Text Transcript from Video
hello frames this video an application
of derivatives part 12 is brought to you
by exam fear calm no more you from exam
before watching this video please make
sure that you have watched part 1 - part
11s a chrome example here you have to
find the slope of the normal please note
here instead of tangent I am finding
slope of normal to this curve and the
curve is X is equal to a cos cube theta
and Y is equal to a sine two theta at
theta is equal to PI by four now if you
know if this is the case my slope of
normal is nothing but minus 1 by dy by
DX correct at theta is equal to PI 1
correct because this is nothing but F
less X so my normal is minus 1 by dy by
DX by where cheetah is going to PI by 3
so I have to find V by V DX
since I know that x and y are dependent
but not directly dependent they are
dependent on theta so they are dependent
so how to find dy by DX in such case so
in such case we first find DX by D theta
and then we find dy by D theta and then
we divide to find dy by DX
if you don't know if you're not
understanding this please watch the DVD
the derivative chapter may we explain
such kind of questions how to find the
derivative of equations where x and y
are indirectly dependent so let's find
both the values DX by D theta and dy by
D is done so if my X is equal to a cos 2
theta my D X by D theta may be nothing
but a constant cos cube will become 3
cos square theta and again cause I have
to differentiate this will become minus
sine theta correct this will be my DX by
DT similarly if my Y is equal
to a sine cube theta you see here Y is
nothing but a sine cube theta so here my
dy by D theta is nothing but ei into
sine cubed theta become 3 sine square
theta and sine theta get able
differentiate this becomes cos theta so
all chain dude if I understand in this
please watch the previous chapter where
we have explained how to find
derivatives so I have DX by DT and dy by
D theta the reason why we found these
was because we wanted to find dy by DX
so dy by DX is nothing but dy by D theta
divided by D X by D 0 and that is
nothing but a into 3 sine squared theta
into cos theta by a into 3 cos square
theta into minus sine theta a a
cancelled C 3 cancel sine squared theta
and sine theta cancel and again cos into
cos square cancel so what I am left is
minus sine theta by cos theta and that
is nothing but minus 10 that is the
value of d-y-by-dx because I need dy by
DX here now the same formula apply slope
of normal is nothing but minus 1 by dy
by DX that is again minus tan theta
where theta is you can do PI by 4 minus
minus cancels so there is cot theta and
cot theta is PI by 4 it's got PI by 4
and chord PI by 4 is nothing but 1 we
know and that is my answer so slow of
normal of this curve at theta is equal
to PI by 4 is 1 correct so what we have
done we had this equation x and y which
are indirectly linked with theta so we
found DX by D theta and dy by DT are
then we found B
why buy these because we wanted to find
slope of Norman Escobar Norman's nothing
but minus 1 by d-y-by-dx
let's take one more example here we have
to find the points at which tangent to
the curve this is palette 2x xx so it is
saying that my tangent is parallel to x
axis that means my tangent slope is Z
how I found this
since my tangent is parallel to x axis
that clearly means my tangent slope is 0
because any line that is parallel to x
axis the slope is always correct if my
slope is infinity that means the line is
parallel to Y axis like this it means
slope is 0 that means the line is
neither do excesses the question see is
the tangent is parallel to x axis so
slope of tangent is 0 that is something
we know now we have to find the point at
which slope is 0
tangent slope is 0 for the scope so my
slope of tangent is nothing but dy by DX
at X is equal to X naught so let's find
by D by DX X cube becomes 3x square when
you say dy by DX 3x square becomes 3
into 2 X 9 X becomes 9 and 7 becomes 0
so this is my dy by DX correct and X is
equal to X naught is my tangent slope
and that is 0 so let us put X equal to X
naught you get 3 X naught square minus 6
X naught minus 9 is equal to 0 let's
take three common so they come three
into X naught square minus 2 X naught
minus 3 is equal to 0 its factorize this
this becomes X naught square minus three
X naught plus X naught this becomes 2 X
naught minus 3 is equal to 0 and this is
nothing but X naught minus 3 into X
naught plus 1 is equal to 0 so if we
solve this X naught is equal to 3 or
minus 1 so I have two possible values of
x naught for which my tangent is
palliative excesses but I'm asked the
points that means I need to find y also
so in case 1 my X naught is equal to 3
alpha and y Y is nothing but X cube that
this 3 cube minus 3 into X square 3 into
3 square minus 9x minus t3 plus 7 plus 7
correct you solve this this comes out to
be this 3q Venturi you cancel 9 degrees
minus 27 plus 27 is minus 20 so my first
point is Z comma minus 3 correct where X
is 3 and Y is minus 20 second is X
naught is minus 1 so here Y will come
out to be X cubed that is minus 1 to the
power P minus 3 X square 3 into minus 1
square minus 9 X 9 into minus 1 plus 7
so it becomes minus 1 this becomes minus
3 correct and this becomes plus mine
that is mine let me write minus 1 plus 3
this becomes plus 9 and then plus 7 so
it is 9 plus 7 is 16 and is minus 3
actually 16 minus 4 is 12 so 12 is 9
point so the next point is minus 1 and
12 so these are my two points and there
is my answer
so what I have done I was told to find
the points at which the tangent is 0
slope of the tangent is 0
I wrote the formula of slope of tangent
correct I got this value as and I
equated that guy to be zero because I'm
told that slope of tangent is zero
I found the probable values of X to be 3
and minus 1 since I was asked the points
that means I was asked the values of X
in mind only X so if I want the value of
y in both the cases and I got the answer
to be 3 comma minus twenty eight minus
one comma joy let us take one more
example this is again a little different
example please note here to find the
point at which the curve this at which
the tangent is parallel to the chord
joining this end this slope of tangent
is given it is not zero in this case now
it has some dollars and again similar to
last question the last question the
slope of tangent was 0 because it was
parallel to x axis in this question if
you see this tangent is parallel to the
car joining this point so this is
something like this I have some curve
and this is God joining this point in
this point correct so if these two are
the points and it makes the line the
question says I have to find a point XY
here see this guy is to come up 0 and
this guy's forgiven let's suppose there
is a car then you can make from this and
the questions that you find a point XY
where this tangent is valid do this card
so if you can find the slope of the car
that will become slope of tangent
correct so first thing is let us find
the slope of the car so if I have two
points like this the slope is nothing
but slope of God it will be nothing but
Y 2 minus if I have two point y 2 y 1 x
2 i X 1 the slope is nothing but Y 2
minus y 1 X 2 minus X 1 correct why we
need to line and if I have x1 y1 and x2
y2 be the two point the slope will be y2
minus y1
by x2 minus x1 so this guy is my x1 y1
this guy is X 2 y so Y 2 is 4 o Y 1 is 0
minus 0 X 2 is 4 and X minus 2 so let's
come forward by 2 that is correct this
is my slope so if this is the slope of
the card I am told that the tangent is
parallel to the chord so I can stay that
slope of tangent is nothing but is equal
to slope of corn and that is it going to
do why because this is my card and this
is my ancient and with a parallel slope
will be equal so I have got slope of my
tangent and slope of my tangent of dy by
DX at X is equal to X naught and this is
equal to 2 I have to find the value of x
naught so dy by DX in this case if Y is
equal to X minus 2 square that is
nothing but if you expand this or dy by
DX is nothing but 2 into X minus 2
correct
so my dy by DX is nothing but 2 into X
minus 2 and this at X is equal to X
naught is equal to du so let me put X
equal to X naught so this becomes 2 into
X naught minus 2 is equal to 2 to cancel
this become 1 so I get X naught is equal
to 1 plus 2 that is so my value of x
naught is 3 if my value of x naught is 3
you have to find value of y naught also
so Y naught is nothing but X naught
minus 2 square correct Y naught is
nothing but X naught minus 2 square this
is nothing but 3 minus 2 square that is
1 square that is 1 so my value of x
naught and y naught for which
this this true is 3 comma 1 correct at 3
comma 1 the tangent to this curve will
be parallel to this God correct thank
you visit exam viacom to watch free
educational videos try free on my debts
get the best quality study materials
study from the best tutors and mentors
and much more
thanks on screen
of derivatives part 12 is brought to you
by exam fear calm no more you from exam
before watching this video please make
sure that you have watched part 1 - part
11s a chrome example here you have to
find the slope of the normal please note
here instead of tangent I am finding
slope of normal to this curve and the
curve is X is equal to a cos cube theta
and Y is equal to a sine two theta at
theta is equal to PI by four now if you
know if this is the case my slope of
normal is nothing but minus 1 by dy by
DX correct at theta is equal to PI 1
correct because this is nothing but F
less X so my normal is minus 1 by dy by
DX by where cheetah is going to PI by 3
so I have to find V by V DX
since I know that x and y are dependent
but not directly dependent they are
dependent on theta so they are dependent
so how to find dy by DX in such case so
in such case we first find DX by D theta
and then we find dy by D theta and then
we divide to find dy by DX
if you don't know if you're not
understanding this please watch the DVD
the derivative chapter may we explain
such kind of questions how to find the
derivative of equations where x and y
are indirectly dependent so let's find
both the values DX by D theta and dy by
D is done so if my X is equal to a cos 2
theta my D X by D theta may be nothing
but a constant cos cube will become 3
cos square theta and again cause I have
to differentiate this will become minus
sine theta correct this will be my DX by
DT similarly if my Y is equal
to a sine cube theta you see here Y is
nothing but a sine cube theta so here my
dy by D theta is nothing but ei into
sine cubed theta become 3 sine square
theta and sine theta get able
differentiate this becomes cos theta so
all chain dude if I understand in this
please watch the previous chapter where
we have explained how to find
derivatives so I have DX by DT and dy by
D theta the reason why we found these
was because we wanted to find dy by DX
so dy by DX is nothing but dy by D theta
divided by D X by D 0 and that is
nothing but a into 3 sine squared theta
into cos theta by a into 3 cos square
theta into minus sine theta a a
cancelled C 3 cancel sine squared theta
and sine theta cancel and again cos into
cos square cancel so what I am left is
minus sine theta by cos theta and that
is nothing but minus 10 that is the
value of d-y-by-dx because I need dy by
DX here now the same formula apply slope
of normal is nothing but minus 1 by dy
by DX that is again minus tan theta
where theta is you can do PI by 4 minus
minus cancels so there is cot theta and
cot theta is PI by 4 it's got PI by 4
and chord PI by 4 is nothing but 1 we
know and that is my answer so slow of
normal of this curve at theta is equal
to PI by 4 is 1 correct so what we have
done we had this equation x and y which
are indirectly linked with theta so we
found DX by D theta and dy by DT are
then we found B
why buy these because we wanted to find
slope of Norman Escobar Norman's nothing
but minus 1 by d-y-by-dx
let's take one more example here we have
to find the points at which tangent to
the curve this is palette 2x xx so it is
saying that my tangent is parallel to x
axis that means my tangent slope is Z
how I found this
since my tangent is parallel to x axis
that clearly means my tangent slope is 0
because any line that is parallel to x
axis the slope is always correct if my
slope is infinity that means the line is
parallel to Y axis like this it means
slope is 0 that means the line is
neither do excesses the question see is
the tangent is parallel to x axis so
slope of tangent is 0 that is something
we know now we have to find the point at
which slope is 0
tangent slope is 0 for the scope so my
slope of tangent is nothing but dy by DX
at X is equal to X naught so let's find
by D by DX X cube becomes 3x square when
you say dy by DX 3x square becomes 3
into 2 X 9 X becomes 9 and 7 becomes 0
so this is my dy by DX correct and X is
equal to X naught is my tangent slope
and that is 0 so let us put X equal to X
naught you get 3 X naught square minus 6
X naught minus 9 is equal to 0 let's
take three common so they come three
into X naught square minus 2 X naught
minus 3 is equal to 0 its factorize this
this becomes X naught square minus three
X naught plus X naught this becomes 2 X
naught minus 3 is equal to 0 and this is
nothing but X naught minus 3 into X
naught plus 1 is equal to 0 so if we
solve this X naught is equal to 3 or
minus 1 so I have two possible values of
x naught for which my tangent is
palliative excesses but I'm asked the
points that means I need to find y also
so in case 1 my X naught is equal to 3
alpha and y Y is nothing but X cube that
this 3 cube minus 3 into X square 3 into
3 square minus 9x minus t3 plus 7 plus 7
correct you solve this this comes out to
be this 3q Venturi you cancel 9 degrees
minus 27 plus 27 is minus 20 so my first
point is Z comma minus 3 correct where X
is 3 and Y is minus 20 second is X
naught is minus 1 so here Y will come
out to be X cubed that is minus 1 to the
power P minus 3 X square 3 into minus 1
square minus 9 X 9 into minus 1 plus 7
so it becomes minus 1 this becomes minus
3 correct and this becomes plus mine
that is mine let me write minus 1 plus 3
this becomes plus 9 and then plus 7 so
it is 9 plus 7 is 16 and is minus 3
actually 16 minus 4 is 12 so 12 is 9
point so the next point is minus 1 and
12 so these are my two points and there
is my answer
so what I have done I was told to find
the points at which the tangent is 0
slope of the tangent is 0
I wrote the formula of slope of tangent
correct I got this value as and I
equated that guy to be zero because I'm
told that slope of tangent is zero
I found the probable values of X to be 3
and minus 1 since I was asked the points
that means I was asked the values of X
in mind only X so if I want the value of
y in both the cases and I got the answer
to be 3 comma minus twenty eight minus
one comma joy let us take one more
example this is again a little different
example please note here to find the
point at which the curve this at which
the tangent is parallel to the chord
joining this end this slope of tangent
is given it is not zero in this case now
it has some dollars and again similar to
last question the last question the
slope of tangent was 0 because it was
parallel to x axis in this question if
you see this tangent is parallel to the
car joining this point so this is
something like this I have some curve
and this is God joining this point in
this point correct so if these two are
the points and it makes the line the
question says I have to find a point XY
here see this guy is to come up 0 and
this guy's forgiven let's suppose there
is a car then you can make from this and
the questions that you find a point XY
where this tangent is valid do this card
so if you can find the slope of the car
that will become slope of tangent
correct so first thing is let us find
the slope of the car so if I have two
points like this the slope is nothing
but slope of God it will be nothing but
Y 2 minus if I have two point y 2 y 1 x
2 i X 1 the slope is nothing but Y 2
minus y 1 X 2 minus X 1 correct why we
need to line and if I have x1 y1 and x2
y2 be the two point the slope will be y2
minus y1
by x2 minus x1 so this guy is my x1 y1
this guy is X 2 y so Y 2 is 4 o Y 1 is 0
minus 0 X 2 is 4 and X minus 2 so let's
come forward by 2 that is correct this
is my slope so if this is the slope of
the card I am told that the tangent is
parallel to the chord so I can stay that
slope of tangent is nothing but is equal
to slope of corn and that is it going to
do why because this is my card and this
is my ancient and with a parallel slope
will be equal so I have got slope of my
tangent and slope of my tangent of dy by
DX at X is equal to X naught and this is
equal to 2 I have to find the value of x
naught so dy by DX in this case if Y is
equal to X minus 2 square that is
nothing but if you expand this or dy by
DX is nothing but 2 into X minus 2
correct
so my dy by DX is nothing but 2 into X
minus 2 and this at X is equal to X
naught is equal to du so let me put X
equal to X naught so this becomes 2 into
X naught minus 2 is equal to 2 to cancel
this become 1 so I get X naught is equal
to 1 plus 2 that is so my value of x
naught is 3 if my value of x naught is 3
you have to find value of y naught also
so Y naught is nothing but X naught
minus 2 square correct Y naught is
nothing but X naught minus 2 square this
is nothing but 3 minus 2 square that is
1 square that is 1 so my value of x
naught and y naught for which
this this true is 3 comma 1 correct at 3
comma 1 the tangent to this curve will
be parallel to this God correct thank
you visit exam viacom to watch free
educational videos try free on my debts
get the best quality study materials
study from the best tutors and mentors
and much more
thanks on screen
About this Video
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Dec 22, 2024 Last updated |
Video Description: Examples : Tangents and Normals for JEE 2024 is part of Mathematics (Maths) Class 12 preparation.
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