history of mathematics - National STEM Centre

14 Answers
6 The equation z 2 + az + b 2 =0 has solutions
z = - -y a ± -^-^ a 2 - b 2 . Both these solutions are always
negative.
Activity 7.6, page 91
\ A method for the solution of all construction
problems, which is generally applicable.
Activity 7.7f page92
1 Choose values yourself for the number of unknown
quantities, to which no equation corresponds.
Activity 7.8, page 92
1 The coordinates of X are (x, y]. As
distance AX = 2 x distance BX, you have
2
+ y = 2x - a) 2 + y 2 which leads to
2 = 4( (x - a) 2 + y2 ) or
3x 2 + 3y 2 - Sax + 4a 2 =0.
2 Write the equation as x~ +y — •|ox + -fa = 0 or as
(jc - 4«) +y 2 =%a 2 . This shows that the curve is a
circle, centre (-| a, 0), radius ^ a.
Activity 7.9, page 92
I Let PA = x and PC = y. This is, in effect, taking n as
the jc-axis and / as the >>-axis. Then, if PA 2 = PB x PC,
then jc 2 = x - a x\y , the modulus signs showing that
the positive value of the distance is always taken. Then
v 2 =- or v: or y = - 7
x — a\
x-a
2 Take a = 1. Then the graph is a hyperbola and its
reflection in the jc-axis.
174
Activity 7.10, page 95
1 Descartes wants to go further, and to distinguish
several other classes of problem within the linear class.
2 In antiquity, the linear curves were defined as
mechanical because they could only be drawn with an
instrument. In Descartes's opinion this is remarkable
because the straight line and the circle need instruments,
the straight edge and compasses. The fact that
instruments are more complicated, and the resulting
curves therefore more inaccurate is, according to
Descartes, no reason to call the curve mechanical.
3 Between any pair of points a straight line always can
be drawn.
Through any point a circle can be drawn with a given
centre.
4 Descartes says that the mathematicians of antiquity
did not restrict themselves to these two postulates. For
instance, they assumed that any given cone can be
intersected by a given plane. He uses the argument here
because he wants to plant an idea which he will use to
back his later suggestion to accept construction curves
other than the line and circle.
5 The curve must be described by a continuous motion
or by a succession of several continuous motions, each
motion completely determined by those which precede it.
Activity 7.11, page 95
1 Q and R are the feet of the perpendiculars from P to
the y-axis and the jc-axis.
y i
Q Pfc >•)
J(
R
RP PA
As triangle PAR is similar to triangle B AO, —— = —— ,
* B OB BA
so y — — yB . Similarly, it follows from the similar
triangles PBQ and ABO that jt = - - JCA .
2 OA 2 + OB 2 = AB 2 so -^ x 2 + ^y 2 = c 2 . Therefore
b a
the coordinates (x,y) of P satisfy the equation
x 2 y 2
— j- + —5- = 1. This is the equation of an ellipse.
b a