Equation of Parabola - Conic Sections, Math, Class 11 Video Lecture

we now define parabola parabola is the
set of all points parabola is the set of
all points that are equidistant from a
fixed line
and a fixed point so the fixed line we
will call directrix and fix it point we
will call focus so let us see this in
the diagram so we have fixed line as the
vertical line and there is a fixed point
on the x-axis and you see as we are
traversing on the parabola we are
equidistant from the fixed point and the
fixed line so you can see this clearly
you are equidistant from the focus fixed
point and fix it line here both the legs
of the point are of equal length so you
can see this clearly now let us call
this point as P and this point which is
the fixed point is called the focus SE
focus lies at point a comma 0 so we have
a fixed line also focus we are denoting
by a and the point B lies on the fixed
line so this fixed line is called the
directrix again the fixed point is
called focus fixed line is called
directrix and point b is minus a gamma Y
point P is X comma Y notice that the
equation of directrix is x equals to
minus a so fixed line is x equals to
minus a so first thing you notice is
that PV is perpendicular to the
directrix so what is the length of the
segment PB this is equal to square root
of x minus minus a that will be X plus a
square plus y minus y square this will
give you X plus a square square root now
according to definition PB is equal to
a so the point on the parabola is
equidistant from the directrix and the
focus so what is pa pa is X minus a
square plus y minus 0 square so you have
X minus a square plus y square so again
by the definition of the pub the
parabola PB is equal to PA the distance
from the fixed headline is equal to
distance from the focus now you square
both sides and this will give you X plus
a square is equal to X minus a square
plus y square or you can say y square is
equal to X plus a square minus X minus a
square now you simplify it we are using
the formula x square minus P Square is X
minus B times X plus B so we get y
square is equal to 2a times 2x which is
for ax so let us summarize what we have
obtained so this point here first we
have to call this point 0 0 as the
vertex of the parabola so 0 0 is the
vertex of the parabola so let us
summarize equation of parabola with the
vertex at the origin and the focus
and the focus at Point a zero and
directrix x equals to minus a is given
as Y square equals 24 ax so equation of
the parabola with vertex at the origin
focus at a zero and directrix x equals
to minus a is given as Y square equals
24 ax