JEE Exam > JEE Videos > Mathematics (Maths) for JEE Main & Advanced > Parabola : Summary and Examples
Parabola : Summary and Examples Video Lecture | Mathematics (Maths) for JEE Main & Advanced
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FAQs on Parabola : Summary and Examples Video Lecture - Mathematics (Maths) for JEE Main & Advanced
| 1. What is a parabola? |  |
Ans. A parabola is a U-shaped curve that can be formed by the graph of a quadratic function. It is symmetric and has a vertex as its highest or lowest point.
| 2. How do you find the vertex of a parabola? | |
Ans. To find the vertex of a parabola in the form y = ax^2 + bx + c, you can use the formula x = -b/2a. This will give you the x-coordinate of the vertex. To find the y-coordinate, substitute the x-coordinate into the equation.
| 3. What are the different forms of the equation of a parabola? | |
Ans. There are three main forms of the equation of a parabola: standard form (y = ax^2 + bx + c), vertex form (y = a(x-h)^2 + k), and intercept form (y = a(x-p)(x-q)). Each form has its own advantages depending on the given information about the parabola.
| 4. How can I determine if a parabola opens upward or downward? | |
Ans. The coefficient of the x^2 term determines the direction of the parabola. If the coefficient is positive, the parabola opens upward, and if it is negative, the parabola opens downward.
| 5. What are some real-life examples of parabolas? | |
Ans. Parabolas can be found in various real-life scenarios such as the trajectory of a thrown object, the shape of satellite dishes, the design of arches in architecture, and the path of water in a fountain.
Text Transcript from Video
hello friends this video conic section
part six is brought to you by
exampie a calm no more fear from exam
before watching this video please make
sure that you have watched coding
section part 1 - bar 5 some observation
on parabola first parabola is symmetric
with x axis if equation has y square now
if equation is x squared term it is
symmetric along Y axis if you see this
is the parabola this equation is y
square is equal to 4x and if this is the
parabola the equation is X square is
going to be for a while so if this is Y
squared term
this is symmetric along x axis if it is
x squared is equal to 4a y term this is
symmetric along Y axis also also when
the extra symmetry is along x axis this
case the parabola opens to write in the
coefficient of x is positive this one
and parabola opens to left in the
coefficient of x is negative you see
this one so here the coefficient of x is
negative so it opens in the left side
similarly when the parabola is symmetric
along Y axis it opens towards upwards if
the wire has positive and it opens
downward if Y is negative this is y
negative Y positive this X negative and
this is a light promise again this is X
positive so the other four cases which
which we observe first thing if it is Y
square is equal to 4 X that means this
is symmetric along x axis either of
these two scenario it is x square as
well to food away
that means this parabola is symmetric
along Y axis if it is positive X X then
this opens towards there if it is
negative x it opens towards right if it
is positive Y it opens toward up if it
is negative Y it opens towards now let
me introduce to a new term called
lattice rectum
later sector parabola is a line segment
perpendicular to the X
parabola and passes through focus and
whose endpoints lies on the parabola you
see this is the focus any line that
passes through focus perpendicular to x
axis this is XS
it should not be like this or like this
it should be perpendicular to axis and
it should meet in the parabola its
endpoints are in the parabola so this
line is called latest rectum
let me repeat line that is pass it into
focus
perpendicular to axis and endpoints are
meeting in the is called lettuce rectum
the length of lettuce victim is for a
we'll derive this so let's suppose this
is the parabola Y square is equal to 4x
and now let us derive the letters rectum
so the focus we know is a comma zero
because this is a pair of parable log
equation y square is equal to 4x this
point we know focus is a comma 0 now we
let's draw this vertex which is not dead
this is the directrix we have and this
is nothing but X is equal to minus 8 and
this is the line let's suppose this is
point P now we are told that and this is
point Q since we know that P Q is equal
to P F because this is a point s focus
this we are told now this length is a
this length is a why because this is
minus a comma 0
this length is a this length is also a
why because a comma 0 this total length
we know that Q P is nothing but 2 a too
small a this length is 2 a this length
and this is the same because these two
lines are parallel QP is 2 a now we know
that QP is equal to bf
you say PF is nothing but is equal to QP
and that this to it so say this is 2a
this is also 2 a so we have this also s
student from here to here now since this
is 2a let's suppose this is P - now PF
is equal to 2 it why because PF is going
to Q piece
so now PF is equal to 2 e 8p - f will
also be equal to do it why because this
is symmetric
this whole thing is symmetrical about x
axis so P does F is equal to do it so we
know that let us like them is nothing
but P B - this whole length from here to
here is called intersect them so this is
nothing but P F plus FP - that is
nothing but 2 a plus 2 a and is equal to
4 a so this is our answer so the length
of lettuce ectomy is equal to 4 it we
have to write this we have drawn is what
directrix and we found that this QP is 2
a so PF is also to a and T it has equal
to 1 in this question we are told that
there is a parabola Y square is equal to
8x that this Y square is equal to 4 into
2x is going to because we have the
formula Y square is equal to 4 ax
correct Y square is it 8x we have to
find the coordinates of focus X this
equation of directors and data sector
since a is equal to 2 here we can plot
this graph like this parabola and we can
say X go to minus 2 because this is
nothing but X is equal to minus a and
this is a comma 0 correct so when a is
equal to 2 we have 2 comma 0 and is
equal to minus 2x squared - -2 so what
we have done first we have first one the
value of a that is 2 and also we have
observed that this equation of the form
Y square is equal to 4 ax and we have
drawn the graph like this because a was
positive
now we can clearly see that the focus is
F is nothing but 2 comma zero correct X
is nothing but X s is this line that
this Y is equal to zero this is the line
and equation of directrix DX that is
nothing but X is equal to minus 2
because X equal to minus a n is 2 and
later sector L is nothing but 4 times a
that is 4 into 2 that is 8 so we have
all this values what we have done very
simple we have found the value of a and
we observe that this equation is of the
form Y square is going to for a X drawn
the graph and we have found the value of
focus exists directrix and latest victim
hello political example of parabola you
find the equation of parabola with
vertex 0 0 and focus 0 comma 2 if you
see here the focus is 0 comma 2 that
means the focus is on the y-axis x is 0
and Y is 2 H 0 the vertex is 0 0 gives
the vertex and the focus is still a
comma 2 this is the focus so the gate
the graph like this is the vertex not
correct in this you know this part this
will be the graph and then here we'll
have directrix this is 0 comma 2 so this
will be 0 comma minus 2 thus we can see
that a is equal to true here he is good
a to him and this opens on the top so
equation is of this form X square is
good to go if I now input a is going to
do we get X square is going to
eight five and this is the occasion of
the crown
what they are done we have plotted the
graph we have found the value of a is
going to do and we have found that the
graphs of the home X squares rental for
a while in this place the value of a is
going to do and got the answer in this
question we had told to find the
equation of parabola this symmetric
about the y-axis and passes through
point two comma minus three
let us draw the graph first this is the
axis x axis this is y axis now it is
told that the parabola is symmetric
about y axis so the parabola can be like
this or the parabola can be like this
both this parabola is symmetric about Y
axis it is also told that this parabola
passes through two comma minus 3 to
minus 3 this will be some point 2 comma
minus V I think so since this parabola
possible 2 comma minus 3 this is not the
graph here doesn't work this is the
graph we are talking about that means
the equation is of this form X square is
equal to minus 4a y correct it also told
that 2 and minus 3 is the value in the
paragraph so 2 a - we will satisfy this
equation so we get 2 square is equal to
minus 4 into a into minus 3 or code is
going to take la or we get a is equal to
4 by 12 that is 1 by 3 so we have got
the value of K see we have got the value
of a with this put the value here so
what we get X square equal to minus 4 is
equal to 1 by 3 into 1 or X square is
equal to minus 4 by 3 this is my
equation of the parabola very simple
what we have done we are just plotted
the graph and then we know that to go -
we possess great so this is the form of
equation we are looking for
you wrote the equation and we have on
the value thank you visit exam Viacom to
watch free educational videos try free
online tests get the best quality study
materials study from the best tutors and
mentors and much more thanks again
part six is brought to you by
exampie a calm no more fear from exam
before watching this video please make
sure that you have watched coding
section part 1 - bar 5 some observation
on parabola first parabola is symmetric
with x axis if equation has y square now
if equation is x squared term it is
symmetric along Y axis if you see this
is the parabola this equation is y
square is equal to 4x and if this is the
parabola the equation is X square is
going to be for a while so if this is Y
squared term
this is symmetric along x axis if it is
x squared is equal to 4a y term this is
symmetric along Y axis also also when
the extra symmetry is along x axis this
case the parabola opens to write in the
coefficient of x is positive this one
and parabola opens to left in the
coefficient of x is negative you see
this one so here the coefficient of x is
negative so it opens in the left side
similarly when the parabola is symmetric
along Y axis it opens towards upwards if
the wire has positive and it opens
downward if Y is negative this is y
negative Y positive this X negative and
this is a light promise again this is X
positive so the other four cases which
which we observe first thing if it is Y
square is equal to 4 X that means this
is symmetric along x axis either of
these two scenario it is x square as
well to food away
that means this parabola is symmetric
along Y axis if it is positive X X then
this opens towards there if it is
negative x it opens towards right if it
is positive Y it opens toward up if it
is negative Y it opens towards now let
me introduce to a new term called
lattice rectum
later sector parabola is a line segment
perpendicular to the X
parabola and passes through focus and
whose endpoints lies on the parabola you
see this is the focus any line that
passes through focus perpendicular to x
axis this is XS
it should not be like this or like this
it should be perpendicular to axis and
it should meet in the parabola its
endpoints are in the parabola so this
line is called latest rectum
let me repeat line that is pass it into
focus
perpendicular to axis and endpoints are
meeting in the is called lettuce rectum
the length of lettuce victim is for a
we'll derive this so let's suppose this
is the parabola Y square is equal to 4x
and now let us derive the letters rectum
so the focus we know is a comma zero
because this is a pair of parable log
equation y square is equal to 4x this
point we know focus is a comma 0 now we
let's draw this vertex which is not dead
this is the directrix we have and this
is nothing but X is equal to minus 8 and
this is the line let's suppose this is
point P now we are told that and this is
point Q since we know that P Q is equal
to P F because this is a point s focus
this we are told now this length is a
this length is a why because this is
minus a comma 0
this length is a this length is also a
why because a comma 0 this total length
we know that Q P is nothing but 2 a too
small a this length is 2 a this length
and this is the same because these two
lines are parallel QP is 2 a now we know
that QP is equal to bf
you say PF is nothing but is equal to QP
and that this to it so say this is 2a
this is also 2 a so we have this also s
student from here to here now since this
is 2a let's suppose this is P - now PF
is equal to 2 it why because PF is going
to Q piece
so now PF is equal to 2 e 8p - f will
also be equal to do it why because this
is symmetric
this whole thing is symmetrical about x
axis so P does F is equal to do it so we
know that let us like them is nothing
but P B - this whole length from here to
here is called intersect them so this is
nothing but P F plus FP - that is
nothing but 2 a plus 2 a and is equal to
4 a so this is our answer so the length
of lettuce ectomy is equal to 4 it we
have to write this we have drawn is what
directrix and we found that this QP is 2
a so PF is also to a and T it has equal
to 1 in this question we are told that
there is a parabola Y square is equal to
8x that this Y square is equal to 4 into
2x is going to because we have the
formula Y square is equal to 4 ax
correct Y square is it 8x we have to
find the coordinates of focus X this
equation of directors and data sector
since a is equal to 2 here we can plot
this graph like this parabola and we can
say X go to minus 2 because this is
nothing but X is equal to minus a and
this is a comma 0 correct so when a is
equal to 2 we have 2 comma 0 and is
equal to minus 2x squared - -2 so what
we have done first we have first one the
value of a that is 2 and also we have
observed that this equation of the form
Y square is equal to 4 ax and we have
drawn the graph like this because a was
positive
now we can clearly see that the focus is
F is nothing but 2 comma zero correct X
is nothing but X s is this line that
this Y is equal to zero this is the line
and equation of directrix DX that is
nothing but X is equal to minus 2
because X equal to minus a n is 2 and
later sector L is nothing but 4 times a
that is 4 into 2 that is 8 so we have
all this values what we have done very
simple we have found the value of a and
we observe that this equation is of the
form Y square is going to for a X drawn
the graph and we have found the value of
focus exists directrix and latest victim
hello political example of parabola you
find the equation of parabola with
vertex 0 0 and focus 0 comma 2 if you
see here the focus is 0 comma 2 that
means the focus is on the y-axis x is 0
and Y is 2 H 0 the vertex is 0 0 gives
the vertex and the focus is still a
comma 2 this is the focus so the gate
the graph like this is the vertex not
correct in this you know this part this
will be the graph and then here we'll
have directrix this is 0 comma 2 so this
will be 0 comma minus 2 thus we can see
that a is equal to true here he is good
a to him and this opens on the top so
equation is of this form X square is
good to go if I now input a is going to
do we get X square is going to
eight five and this is the occasion of
the crown
what they are done we have plotted the
graph we have found the value of a is
going to do and we have found that the
graphs of the home X squares rental for
a while in this place the value of a is
going to do and got the answer in this
question we had told to find the
equation of parabola this symmetric
about the y-axis and passes through
point two comma minus three
let us draw the graph first this is the
axis x axis this is y axis now it is
told that the parabola is symmetric
about y axis so the parabola can be like
this or the parabola can be like this
both this parabola is symmetric about Y
axis it is also told that this parabola
passes through two comma minus 3 to
minus 3 this will be some point 2 comma
minus V I think so since this parabola
possible 2 comma minus 3 this is not the
graph here doesn't work this is the
graph we are talking about that means
the equation is of this form X square is
equal to minus 4a y correct it also told
that 2 and minus 3 is the value in the
paragraph so 2 a - we will satisfy this
equation so we get 2 square is equal to
minus 4 into a into minus 3 or code is
going to take la or we get a is equal to
4 by 12 that is 1 by 3 so we have got
the value of K see we have got the value
of a with this put the value here so
what we get X square equal to minus 4 is
equal to 1 by 3 into 1 or X square is
equal to minus 4 by 3 this is my
equation of the parabola very simple
what we have done we are just plotted
the graph and then we know that to go -
we possess great so this is the form of
equation we are looking for
you wrote the equation and we have on
the value thank you visit exam Viacom to
watch free educational videos try free
online tests get the best quality study
materials study from the best tutors and
mentors and much more thanks again
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About this Video
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Oct 23, 2024 Last updated |
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