The document discusses various conic sections including circles, parabolas, ellipses, and hyperbolas. It provides definitions, key properties, and standard equations for each type of conic section. Examples of applications in fields like planetary motion and antenna design are also mentioned. Key aspects like focus, directrix, eccentricity, vertex and axis are defined for parabolas, ellipses, and hyperbolas. Methods to find characteristics like latus rectum, equation from geometric properties are also demonstrated.

2. CONIC SECTION IS A STUDY OF CURVES LIKE
CIRCLES, PARABOLAS, HYPERBOLA, ELLIPSE
.THIS CURVES ARE COMMONLY KNOWN AS
CONIC SECTIONS OR CONICS .BECAUSE THEY
CAN BE OBTAINED AS INTERSECTION OF A
PLANE WITH A DOUBLE NAPPED RIGHT
CIRCULAR CONE . THESE CURVES HAVE A
VERY WIDE RANGE OF APPLICATIONS IN
FIELDS SUCH AS PLANETRY MOTION, DESIGN
OF TELESCOPE AND ANTENNAS, REFLECTOR
ETC………………………

3. +-------------------------+-------------------------+--------------------------+--------------------------+
DEFINATION 1: A CIRCLE IS THE SET OF ALL POINTS SUCH THAT
THEY ALL ARE EQUIDISTANT FROM A FIXED POINT.
DEFINATION 2: A CIRCLE IS LOCUS PF A POINT WHICH MOVES IN
SUCH A WAY THAT IT SDISTANCE FROM A FIXED POINT REMAINS
CONSTANT.
THE FIXED POINT IS CALLED AS CENTER OF THE CIRCLE.
THE DISTANCE BETWEEN THE MOVING POINT AND THE FIXED
POINT IS CALLED RADIUS.
TWICE THIS RADIUS GIVES DIAMETER.
CIRCLE
CENTER
RADIUS

4. Y
P(X,Y)
X
C
O

5. ᶩ
B3
B2
B1
• F
P2
P1
P3
DIRECTRIX
P1F=P1B1
P2F=P2B2
P3F=P3B3
PARA BOLA
MEANS
FOR THROWING
i.e THE SHAPE DESCRIBED WHEN
YOU THROW A BALL IN AIR.
DEFINATION : A PARABOLA IS SET OF
ALL POINTS IN A PLANE THAT ARE
EQUIDISTANT FROM A FIXED LINE
AND A FIXED POINT IN THE PLANE.
THE FIXED LINE IS CALLED THE
IRECTRIX OF THE PARABOLA AND
THE FIXED POINT F IS CALLED THE
FOCUS. A LINE THROUGH THE FOCUS
AND PERPENDICULAR TO THE
DIRECTRIX IS CALLED THE AXIS OF
THE PARABOLA.THE POINT OF THE
INTERSECTION OF PARABOLA WITH
THE AXIS IS CALLLED THE VERTEX OF
THE PARABOLA.

6. DIRECTRIX
VERTEX
AXIS
PARABOLA IS THE LOCUS OF THE POINTS SUCH
THAT ITS DISTANCE FROM A FIXED POINT AND
FIXED LINE IS SAME.
EQUATION OF PARABOLA =
Y2=4ax

7. Y2=4ax
Y’
X’ X
Y
O
F(a,O)
.
Y’
X’
Y
OF(-a,O)
. X
Y2=-4ax

8. X2=4ay
X’
X’ X
X
F(O, a)
F(O,-a)
Y’
Y
Y’
Y
.
.
X2=-4ay
EQUATION
OF
PARABOLA

9. SOME OBSERVATIONS ON STANDARD EQUATION OF
PARABOLAS:-
•Parabola is symmetric with respect to the axis of the
parabola. If the equation has a y2 term, then the axis
of the symmetry is along the y-axis.
•When the axis of symmetry is along the x-axis the
parabola opens to the :=
Right if the coefficient of x is positive
Left if the coefficient of x is negative.
•When the axis of symmetry is along the y-axis the
parabola opens to the :=
Upward if the coefficient of y is positive,
Downwards if the coefficient of y is negative.

10. LATUS RECTUM
DEFINATION : LATUS RECTUM OF A PARABOLA
IS A LINE SEGMANT PERPENDICULAR TO THE
AXIS OF THE PARABOLA THROUGH THE FOCUS
AND WHOSE END POINTS LIE ON THE
PARABOLA,THROUGH THE FOCUS AND WHOSE
END POINTS LIE ON THE PARABOLA.

11. Y
XM O
.
F(a,0)
AC
B
BY DEFINATION AC=FM=2a
AF=2a
And since the parabola is symmetric with respect to x
axis AF=FB and so
AB=Length of LATUS RECTUM=4a
HOW TO FIND THE LENGTH OF LATUS RECTUM ?

12. DEFINATION:- A HYPERBOLA IS THE SET OF
ALL POINTS IN THE PLANE,THE DIFFERENCE OF
WHOSE DISTANCES FROM THE FIXED POINTS IN
THE PLANE IS CONST. THE TERM DIFFERENCE IS
EQUAL TO THE FARTHER POINT MINUS THE
CLOSER POINT .
HYPERBOLA IS FORMED WHEN THE PLANE CUTS
NAPPES AND THE CURVES OF INTERSECTION OF
THE CONE IS HYPERBOLA.

13. FOCUS FOCUS
CONJUGATE AXIS
TRANSVERSE AXIS
CENTER
VERTEX
. .F1 F2
HYPERBOLA
P1F2-P1F1 = P2F2-P2F1 = P3F1-P3F2
P1
P2P3
0
VERTEX

14. BA
a
b
2a
2c c
Y
Y1
X1
XF1
F2
. .
WE DENOTE THE DISTANCE BETWEEN TWO FOCI TO BE 2c
AND THE DISTANCE BETWEEN TWO VERTICES BY 2a.
THERFORE WE DEFINE QUANTITTY “b” AS
b=
OR
√C
2
-A
2
b
2
=c
2
-a
2
ALSO 2b IS THE LENGTH OF CONJUGATE AXIS

15. DEFINATION OF CONIC: A CONIC IS THE LOCUS OF A POINT (P) WHICH MOVES SO THAT ITS DISTANCE
FROM A FIXED POINT (F) IS IN CONATANT RATIO (e) TO ITS PERPENDICULAR DISTANCE FROM A FIXED
STRAIGHT LINE (d).
i.e.
(CONSTANT)=Locus of P is conic
THE FIXED POINT F IS CALLED FOCUS; LINE “d” IX CALLED DIRECTRIX AND E IS CALLED ECCENTRICITY OF THE
CONIC.|PM| IS PERPENDICULAR DISTANCE OF P FROM DIRECTRIX.
FOR e=1, CONIC IS CALLED PARABOLA.
FOR 0<e<1,CONIC IS CALLED ELLIPSE.
FOR e>1, CONIC IS CALLED HYPERBOLA.
EQUATION OF STANDARD PARABOLA IS y2=4ax
EQUATION OF STANDARD ELLLIPSE (x2/a2)+(y2/b2)
EQUATION OF STANDARD HYPERBOLA (x2/a2)-(y2/b2)

17. •FOR WHAT POINT ON THE PARABOLA y2=16x IS THE ABSCISSA
EQUAL TO TWICE THE ORDINATE.
•FIND THE ECCENTRITY OF HYPERBOLA WITH A FOCI ON x-AXIS IF
THE LENGTH OF ITS CONJUGATE AXIS IS 3/4 OF THE LENGTH OF ITS
TRANSVERSE AXIS .
•FIND THE EQUATION OF THE CIRCLE DRAWN ON A DIAGONAL OF
RECTANGULAR AS ITS DIAMATER WHOSE SIDES ARE GIVEN BY
x=6,x=-3,y=3,y=-1.
•FOR WHAT VALUE OF k,ARE THE POINTS (2,0),(0,0),(0,2),AND (2,k)
ARE CONCYCLIC.
•FIND THE EQUATION OF THE CIRCLE WHOSE AREA IS 154 sq units
AND HAVING 2x-3y=0 AND x+4y-5=0 AS EQUATION OF DIAMETERS.

18. The given lines are
2x-3y+12=0 …(1)
x+4y-5=0 …(2)
Since lines (1) & (2) are diameters, the center of the given
circle is the inter section of these two diameters.
On solving (1) & (2), we get
x=-3 and y=2
Therefore center is (-3,2).Let ‘r’ be the radius of circle
area=∏r2 = 154
=49 ,r =7.
Equation of circle is
r2
(x+3)2+(y-2)2=72
x2+y2+6x-4y-36=0

19. Given,
the equation of the parabola is
=16x …..(1)
Let P(x,y) be a point on the parabola
Given
x=2y
y2=16(2y)
y2=32y
y(y-32)=0
y=0 or 32
x=0 or 64 …….since x=2y
Therefore
Point=
(0,0) or(32,64)
(ANS)
y2

20. Let foci of the hyperbola are on x-axis, so let the equation of
hyperbola be
=1 where a,b>0
Transverse axis =2a,Conjugate axis =2b.
Now given conjugate axis =3/4 transverse axis
2b= (2a)
4b=3a b= a
We know
x2/a2+ y2/b2
b2=a2(e2-1)
b2=a2e2-a2
a2e2=b2+a2
e=5a/4a since we know b=3a/4
e=
(ANS)

21. The given sides of rectangle are
x=6, ….(1)
x=3, ….(2)
y=3, ….(3)
y=-1 ….(4)
The sides (1) and (3) intersects at the point a(6,3) and sides
(2) and (4) intersects at point c(-3,-1).the equation of circle
with diagonal AC as diameter is
D x=3 C
y=3 y= -1
A x=6 B

22. (x-6)(x+3)+(y-3)(y+1)=0
x2-3x+8+y2-2y-3=0
x2+y2-3x-2y+5=0
Hence the equation f the circle is written above
Let the equation of the circle passing through the three points
(2,0),(0,0),(0,2) be
………(1)
Substitute the points (2,0),(0,0),(0,2),(2,k) in the equation (1), we get
4+4g+c=0;c=0
4+4f+c=0
On solving these equations, we get
C=0 and g=f=1
substituting these values of g,f,c in equation(1)
X2+y2-2x-2y+5=0
x2+y2+2gx+2fy+c=0

23. Since these given points are concyclic
(2,k) satisfies this equation
4+k-4-2k=0
k2-2k=0
K=0,2
(ans)

25. 1. Show that the equation 3y2-10x-12y-18=0
represents a parabola. Find its vertex, focus and
directrix.
2. Find the vertex, axis, focus, latus rectum and
directrix of the parabola X2-3x+2y+5=0 also draw its
rough sketch.
3. A double ordinate is of length 8a.prove that lines
form the vertex to two end are at right angles.
4. Find the equation of a hyperbola in the standard
form having distance between directrix 4√3/3 and
passing through the point(2,1)
5. Find the length of intercepts of the circle X2+y2+8x-
4y+2=0 on the coordinate axes.

26. 1.vertex:(0,0);Focus=(5/6,0);Directrix:x+(5/6)=0
2.vertex=((3/2),(-11/8)),Axis=x-(3/2)=0,Focus=(3/2,-15/8);
LatusRectum=2units
4.3x2-2y2=10;x2-2y2=2
5.2√14