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Distance, Circles, Quadratic Equations

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Calculus and Analytical Geometry
MTH101
LECTUER ­ 5
Distance, Circles, Quadratic Equations
In this lecture we shall discuss:
Exampl
Since we know that if A and B are points
e
on a coordinate line with coordinate a and
b,
respectively, then the distance between A
and B is
Since
So
If x, y and z are base, perpendicular and
The Midpoint Formula
hypotenuse of a right triangle respectively,
then
To derive the midpoint formula, we shall
start with two points on a coordinate line.
In the following figure, the distance between
a and b, and their midpoint is shown.
So, the midpoint is
Now from above figure, the length of the
horizontal side is [x2 - x1 ]  and the length
of the vertical side is  [y2 - y1 ] , so it
follows
from the Theorem of Pythagoras that
Since
So
Mth101
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Calculus and Analytical Geometry
Distance, Circles, Quadratic Equations
Finding the Center and Radius
Theorem:
of a Circle
The midpoint of the line segment joining
two points (x1 , y1 )  and (x2 , y2 ) in a
Example:
coordinate plane is
1
1
midpo int = (  x, y ) = ⎜ (  x1 + x2 )  , ( y1 + y2 )  ⎟
2
2
So comparing with general equation of
circle we find that center=(4,-1) and
radius = 3
An equation of the form
is called a quadratic equation.
at a fixed
The graph of y = ax2 + bx+c
Let
The vertex is the low point on the curve
And
if a>0, as shown.
Then
Example:
Mth101
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Calculus and Analytical Geometry
Distance, Circles, Quadratic Equations
The vertex is the high point on the curve
if a<0, as shown.
Example
:
Sketch the graph of
Thus by using quadratic formula, we have
Mth101
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