Diapositiva 1

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The Circle
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The curves of the second order are called Conic
Sections. The general equation of a conic section
in implicit form is ax²bxycy²dxeyf0
We can see that there are six terms of which
three terms are second order (x², xy and y²), two
terms are first order (x and y) and a constant
(f).
If the determinant is gt0, the conic section is an
Ellipse if the determinant is 0, the conic
section is a Parabola if the determinant is lt0,
the conic section is a Hyperbola. The circle is a
particular ellipse with equal axes.
Remembering that the Locus is a geometric figure
the points of which have all the same property,
so the circle is defined as the locus of the
points P of the cartesian plane which are
equidistant from a fixed point called the
centre. The constant distance r is called the
radius of the circle, while C is the centre of
the circle.
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Now, we want to find the equation of the
circle Let C(xc yc) be the centre of the
circle, let r be the radius of the circle and
P(xy) any point on its circumference.
Using the definition of the locus, the distance
PCr, so we can write
(x - xc) ² (y - yc) ² r ²
squaring both sides
x² xc² - 2xxc y² yc² - 2yyc - r² 0
expanding we have
x² y² - 2xxc - 2yyc xc² yc² - r² 0
and reordering the equation we have
-2xc a -2yc b xc² yc² - r² c
Now if we replace
the equation may be written as x² y² ax
by c 0 which represents the general equation
of a circle
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It is plain that if we have the coordinates of
the centre and the radius of the circle, we can
find the equation through the definition of
locus (x xc)² (y yc)² r² It is plain also
that if we have the equation of the circle, we
can find the centre and the radius to draw the
circumference.
We can see that in the equation of a circle there
are three constants a, b, c, so, if we want to
find the equation of a circle we need three
indipendent conditions to determine the values
of the constants, that is one condition for each
constant. For example we can have the following
conditions
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in this case we equal this expression of r to the
value of the radius
3) the coordinates of a point for which the
circle passes through in this case the general
equation of a circle x² y² ax by c 0
must be satisfied by putting the coordinates of
the point in x and in y of the equation of the
circle
4) in addition we can have the equation of a
straight line which passes through the
centre of the circle in this case we put the
general coordinates of the centre
in x and in y of the equation of the straight
line.
So we put together the three conditions and we
solve this system of three equations
for the variables a, b and c
Now we study some particular circles, beginning
from the general equation
x² y² ax by c 0
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7) If the circle is tangential to the axes, then
the distance between the centre C and the
axes equals the radius, therefore, we can write
xc yc r gt
r
and so
²
c
( ) ²
squaring both sides
( ) ²
gt
for example the equation x² y² 4x 4y 4
0 represents a circle with the centre C (-2-2),
radius r 2 and its tangential the axes
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x² ( mxk )² ax b ( mxk ) c 0
expanding we obtain x² m²x² k² 2kmx ax
bmx bk c 0
and grouping the terms we obtain ( 1 m ) x²
( 2km a bm ) x ( k² bk c ) 0
which represents a quadratic equation that may
have the discriminant
? gt 0 ? 0 ? lt 0
If ? gt 0 the equation has two real distinct
roots, therefore the straight line is SECANT If
? 0 the equation has two real coincident roots,
therefore the straight line is TANGENT If ? lt 0
the equation has two imaginary roots, therefore
the straight line is EXTERNAL.
Now, we want to determine the equations of the
two tangents to a circle from a given point
and these two tangents will be real if the point
P is outside the circle, coincident if the point
P is on the circumference, and imaginary if the
point P is within the circle
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If the point P is outside the circle (Fig. A),
there are two methods to find the equations of
the tangents and that is
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This RADICAL AXIS can be considered as a
particular circle having an infinite radius, and
can be called DEGENERATE CIRCLE.
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This lesson has been made by the teacher Iacino
Serenella for the Liceo Scientifico Statale
Newton of Rome.
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• BIBLIOGRAFIA
• Treatise on conic sections
• by Charles Smith
• of College of Cambridge