THE ADYYAN SCHOOL

ART
INTEGRATED
PROJECT OF
MATHEMATICS SUBMITTED TO- Mr. Tarun
Sir
SUBMITTED BY- Divyanshi
 CONIC SECTION
WHAT IS CONIC SECTION?
A Conic Section is the locus of a point which moves in a plane in such a way that
the ratio of its distance from a fixed point and a fixed line is a constant. Then
● The fixed point is called Focus and is denoted by S.
● The fixed straight line is called Directrix.
● The constant ratio is called Eccentricitt and is denoted by ‘e’.
● The straight line passing through the focus and perpendicular to the directrix
is called the axis of the conic.
● The point of intersection of the conic and it's axis, is called vertex of the conic.
● A line perpendicular to the axis of the conic and passing through its focus is
called latus rectum.
● The point which bisects every chord of the conic passing through it, is called
the centre of the conic.
 DEPENDING ON THE ECCENTRICITY THERE
ARE DIFFERENT CASES THESE ARE:
● Circle when eccentricitt is zero ● Parabola when eccentricity is 1
then the conic is a circle. then the conic is a parabola.
 ● Hyperbola when eccentricity is greater than 1
● Ellipse When eccentricity is less then the conic is a hyperbola.
than 1 then the conic is an
ellipse.
 CIRCLE
A Circle is defined as the locus of a point in a plane, which
moves in such a way that it's distance from a fixed point in that
plane is always constant from a fixed point.
● The fixed point is called the centre of the circle
● The constant distance from the centre is called the radius of
the circle.
● STANDARD EQUATION OF A CIRCLE:
(x-h)^2 + (y-k) ^2 = r^2
 PARABOLA
A parabola is the locus of a point which moves in a place such that it's distance from a fixed point is always equal
to its distance from a straight line in the same.

● The fixed line is called the directrix.

● The fixed point is called the focus of the parabola.
● A line through the focus and perpendicular to the directrix is called the axis of the parabola.

● Point of intersection of intersection of parabola with the axis is called the vertex of the parabola.

● Types of Parabola:
MAIN FACTS ABOUT THOSE FOUR
PARABOLA
PARABOLA VERTEX FOCUS LATUS CO-ORDIN AXIS DIRECTRIX SYMMETR
RECTUM ATE OF L•R Y

Y^2=4ax (0, 0) (a, 0) 4a (a, +2a) Y=0 X=-a X-AXIS

Y^2=-4ax (0, 0) (-a, 0) 4a (-a, +2a) Y=0 X=a X-AXIS

X^2=4ay (0, 0) (0, a) 4a (+2a, a) X=0 Y=-a Y-AXIS

X^2=-4ay (0, 0) (0, -a) 4a (+2a, -a) X=0 Y=a Y-AXIS

ELLIPSE
An ellipse is the set of all points in a plane, the sum of whose distance from two fixed points in the plane is a constant.
● Two fixed points are called foci. Denoted by S1 and S2.
● The mid-point of the line-segment joining the foci, is called the centre of ellipse.
● The line segment through the foci of the ellipse is called major axis. It is denoted by 2a.
● The line segment through the centre and perpendicular to the major axis is called minor axis. It is denoted by 2b.
● The end points of the major axis are called the vertices of the ellipse.
● The eccentricity of the ratio of the distance from the centre of the ellipse to one of the foci and to one of the
vertices of the ellipse. It is denoted by ‘e’.
● Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end
points lie on the ellipse.
● STANDARD EQUATION OF AN ELLIPSE:
HORIZONTAL- x²/a²+y²/b²=1, a>b
VERTICAL- x²/b²+y²/a²=1, a>b
FACTS ABOUT STANDARD ELLIPSE:
Centre (0, 0) (0, 0)

Vertices (+a, 0) (0, +a)

Major axis 2a 2a

Minor axis 2b 2b

Value of c C=√a²-b² C=√a²-b²

Eq. Of major axis Y=0 X=0

Eq. Of minor axis X=0 Y=0

Directrix X=+a²/c or +a/e Y=+b²/c or +b²/ae

Foci (+c, 0) or (ae, 0) (0, +c) or (0, +ae)

Eccentricity e=c/a=√1-b²/a² e=c/a=√1-b²/a²

Lenght of latus rectum 2b²/a 2b²/a

Co-ordinate of latus rectum (+c, +b²/a) (+b²/a, +c)

HYPERBOLA
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

● The two fixed points are called the foci and denoted by S1 and S2.
● The midpoint of the line segment joining the foci, is called centre of hyperbola.
● The line through the foci is called transverse axis.
● The line through the centre and perpendicular to the transverse axis is called conjugate axis.
● The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola. The distance
between two vertices is 2a.
● Eccentricity of the hyperbola is the ratio of the distance of any focus from the centre and the distance of any vertex from the
centre and it is denoted by e.
● It is a line perpendicular to the transverse axis and cuts it at a distance of a²/c from the centre.
● It is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.
● STANDARD EQUATION OF HYPERBOLA:
FACTS ABOUT TWO STANDARD HYPERBOLAS:

Hyperbola Conjugate hyperbola

Transverse axis 2a 2a

Conjugate axis 2b 2b

Value of c C=√a²+b² c=√a²+b²

Vertices (+a, 0) (0, +a)

Directrices x=+a²/c or +a/e y=+a²/c or +a/e

Foci (+ae, 0) or (+c, 0) (0, +ae) or (0, +c)

Eccentricity e=√1+b²/a² or c/a e=√1+b²/a² or c/a

Lenght of latus rectum 2b/a 2b²/a

THANKYOU
-DIVYANSHI