TP

1. CONIC SECTIONS
MADE BY-Aditya Janardhanan
Grade-XI-A
Roll No.-8

2. Conic Sections - Definition
• A conic section is a
curve formed by
intersecting cone with
a plane.
• There are four types
of Conic sections.
1.Circle,2.ellipse,
3.parabola,4.hyperbola.

3. Conic Sections - Four Types

4. PARABOLA

5. What’s in a Parabola
• A parabola is the set of all points in a plane such
that each point in the set is equidistant from a
line called the directrix and a fixed point called
the focus.
(x, y)

6. Parabola
• The Standard Form of a Parabola that opens to the right
and has a vertex at (0,0) is……
)0(42
>= aaxy

7. Parabola
• The Standard Form of a Parabola that opens to the left
and has a vertex at (0,0) is……
)0(42
>−= aaxy

8. Parabola
• The Standard Form of a Parabola that opens up and
has a vertex at (0,0) is……
ayx 42
=

9. Parabola
• The Standard Form of a Parabola that opens down and
has a vertex at (0,0) is……
ayx 42
−=

10. Parabola
• The Standard Form of a Parabola
that opens to the left and has a
vertex at (h,k) is……
)(4)( 2
hxaky −−=−

11. Parabola
• The Standard Form of a Parabola that opens up and has
a vertex at (h,k) is……
)(4)( 2
kyahx −=−

12. Parabola
• The Standard Form of a Parabola that opens
down and has a vertex at (h,k) is……
)(4)( 2
kyahx −−=−

13. Ellipse

14. What is in an Ellipse?
• The set of all points in the plane, the sum of whose
distances from two fixed points, called the foci, is a
constant. (“Foci” is the plural of “focus”, and is
pronounced FOH-sigh.)

15. Ellipse
• The standard form of the ellipse with a center at (0,0)
and a horizontal axis is……
12
2
2
2
=+
b
y
a
x

16. Ellipse
• The ellipse with a center at (0,0) and a horizontal axis
has the following characteristics……
• Vertices ( a,0)
• Co-Vertices (0, b)
• Foci ( c,0)
±
±
±
1
916
22
=+
yx

17. Ellipse
• The standard form of the ellipse with a center at (0,0)
and a vertical axis is……
12
2
2
2
=+
a
y
b
x

18. Ellipse
• The ellipse with a center at (0,0) and a vertical axis has
the following characteristics……
• Vertices (0, a)
• Co-Vertices ( b,0)
• Foci (0, c)
±
±
±
1
819
22
=+
yx

19. Ellipse
• The standard form of the ellipse with a center at (h,k)
and a horizontal axis is……
1
)()(
2
2
2
2
=
−
+
−
b
ky
a
hx

20. Ellipse
• The ellipse with a center at (h,k) and a horizontal axis
has the following characteristics……
• Vertices (h a , k)
• Co-Vertices (h, k b)
• Foci (h c , k)
±
±
±

21. Ellipse
• The standard form of the ellipse with a center at
(h,k) and a vertical axis is……
1
)()(
2
2
2
2
=
−
+
−
a
ky
b
hx

22. Ellipse
• The ellipse with a center at (h,k) and a vertical axis
has the following characteristics……
• Vertices (h, k a)
• Co-Vertices (h b , k)
• Foci (h, k c)
±
±
±

23. Hyperbola
The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does
the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

24. What is a Hyperbola?
• The set of all points in the plane, the
difference of whose distances from two fixed
points, called the foci, remains constant.

25. Hyperbola
• The standard form of the Hyperbola with a center at
(0,0) and a horizontal axis is……
12
2
2
2
=−
b
y
a
x

26. Hyperbola
• The Hyperbola with a center at (0,0) and a horizontal
axis has the following characteristics……
• Vertices ( a,0)
• Foci ( c,0)
• Asymptotes:
x
a
b
y ±=
±
±

27. Hyperbola
• The standard form of the Hyperbola with a center at
(0,0) and a vertical axis is……
12
2
2
2
=−
b
x
a
y

28. Hyperbola
• The Hyperbola with a center at (0,0) and a vertical
axis has the following characteristics……
• Vertices (0, a)
• Foci ( 0, c)
• Asymptotes: x
b
a
y ±=
±
±

29. Hyperbola
• The standard form of the Hyperbola with a center at
(h,k) and a horizontal axis is……
1
)()(
2
2
2
2
=
−
−
−
b
ky
a
hx

30. Hyperbola
• The Hyperbola with a center at (h,k) and a horizontal
axis has the following characteristics……
• Vertices (h a, k)
• Foci (h c, k )
• Asymptotes:
)( hx
a
b
ky −±=
±
±

31. Hyperbola
• The standard form of the Hyperbola with a center at
(h,k) and a vertical axis is……
1
)()(
2
2
2
2
=
−
−
−
b
hx
a
ky

32. Hyperbola
• The Hyperbola with a center at (h,k) and a vertical
axis has the following characteristics……
• Vertices (h, k a)
• Foci (h, k c)
• Asymptotes: )( hx
b
a
ky −±=
±
±

33. Circle

34. What is a circle ?
• A line forming a closed loop,every point on
which is a fixed distance from a centre
point.

35. Circle
• The Standard Form of a circle with a center at (0,0) and a
radius, r, is……..
222
ryx =+
center (0,0)
radius = 2

36. Circles
• The Standard Form of a circle with a center at (h,k) and
a radius, r, is……..
222
)()( rkyhx =−+−
center (3,3)
radius = 2