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Conic Section Ecat

This document provides a summary of key concepts related to conic sections and analytical geometry, specifically focusing on the equations of circles and their properties. It outlines standard and general forms of circle equations, conditions for tangents and normals, and identifies different types of conics based on their general equations. The document also describes the relationships between coefficients and the resulting conic shapes such as circles, parabolas, ellipses, and hyperbolas.

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35 views8 pages

Conic Section Ecat

This document provides a summary of key concepts related to conic sections and analytical geometry, specifically focusing on the equations of circles and their properties. It outlines standard and general forms of circle equations, conditions for tangents and normals, and identifies different types of conics based on their general equations. The document also describes the relationships between coefficients and the resulting conic shapes such as circles, parabolas, ellipses, and hyperbolas.

Uploaded by

abdulsami0548
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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CONIC SECTION CHAPTER 6 2ND YEAR

INTRODUCTION TO ANALYTICAL
GEOMETRY 2ND YEAR CHAPTER 4
SUMMARY

(1)Equation of the circle in a standard form is (x - h) + ( y - 2

k) =r .
2 2

(i) Centre (h, k) (ii) radius ‘ r ’


Equation of a circle in general form is x + y + 2gx + 2 fy
(2) 2 2

+c=0.
(i) centre (-g, - f ) (ii) radius = √[g + f – c] 2 2

(3) The circle through the intersection of the line lx + my + n


= 0 and the circle

my + n) = 0, λ ∈ R1 .
x2 + y + 2gx + 2 fy + c = 0 is x + y + 2gx + 2 fy + c + λ(lx +
2 2 2

(4) Equation of a circle with (x , y ) and (x , y ) as extremit


1 1 2 2

ies of one of the diameters is


(x - x )(x - x ) + ( y - y )( y - y ) = 0 .
1 2 1 2

(5) Equation of tangent at (x , y ) on circle x + y + 2gx +


1 1
2 2

2fy + c = 0 is
xx1 + yy + g(x + x ) + f ( y + y ) + c = 0
1 1 1

(6) Equation of normal at (x , y ) on circle x + y + 2gx +


1 1
2 2

2 fy + c = 0 is
yx1 - xy + g( y - y ) - f (x - x ) = 0 .
1 1 1
Tangent and normal

Condition for the sine y = mx + c to be a tangent to the Conics


Parametric forms

Identifying the conic from the general equation of


conic Ax + Bxy + Cy + Dx + Ey + F = 0
2 2

The graph of the second degree equation is one of a circle, parabola, an ellipse,
a hyperbola, a point, an empty set, a single line or a pair of lines. When,

(1) A = C = 1, B = 0, D = -2h, E = -2k, F = h2 + k 2 - r 2 the general equation


reduces to (x - h)2 + ( y - k )2 = r2 , which is a circle.

(2) B = 0 and either A or C = 0 , the general equation yields a parabola under st


udy, at this level.
(3) A ≠ C and A and C are of the same sign the general equation yields an elli
pse.
(4) A ≠ C and A and C are of opposite signs the general equation yields a hype
rbola

(5) A = C and B = D = E = F = 0 , the general equation yields a point x2 + y2 =


0.
(6) A = C = F and B = D = E = 0 , the general equation yields an empty
set x2 + y2 +1 = 0 , as there is no real solution.
(7) A ≠ 0 or C ≠ 0 and others are zeros, the general equation yield coordinate
axes.
(8) A = -C and rests are zero, the general equation yields a pair of lines x2 - y2
=0.

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