1.Algebra Booster

4.14 Algebra Booster
(vii) We consider two fixed points A(z 1
), B(z 2
) and P(z) is
a variable point moving in such a way that the angle
subtended by segment on the moving point P is a
constant (a).
Pz ()
IM()
z
Pz ()
Az ( 1)
Bz ( )
2
Re()
z
a
Az ( 1)
Bz ( 2)
2
Thus, Ê z - z
ia
Á
ˆ = r ¥ e
Ë z1
- z˜
, where a is a parameter,
¯
whose value depends upon the position of P.
Therefore, the equation satisfied by all the complex
number z lying on the arc AB of a circle can be written
Ê 2
as arg z - z ˆ
Á = a
Ë z - z˜
, where a π 0, p
¯
5. Ellipse
1
If |z – z 1
| + |z – z 2
| = 2a, where 2a > |z 1
– z 2
|, the point z
describes an ellipse having foci at z 1
and z 2
, respectively and
a ΠR + .
Im()
z
Pz ()
Az ( 1)
Bz ( 2)
Re()
z
7. Inverse Point with Respect to a Circle
Two points P and Q are said to be inverse with respect to a
circle with the centre O and the radius r, if
(i) The points O, P, Q are collinear and on the same side of
O.
(ii) OP ◊ OQ = r 2
Note Two points z 1
and z 2
will be the inverse points with
respect to the circle z◊ z + a ◊ z + a ◊ z + r = 0 if and only
if
zz 1 2 + a ◊ z1+ a ◊ z2 + r=
0
26. PTOLEMY’S THEOREM
It states that the product of the lengths of the diagonal of a
convex quadrilateral inscribed in a circle is equal to the sum
of the lengths of the two pairs of its opposite sides.
Dz ( 4)
Cz ( )
3
6. Hyperbola
If |z – z 1
| – |z – z 2
| = 2a, where 2a < |z 1
– z 2
|, the point z describes
a hyperbola having foci at z 1
and z 2
respectively and
a ΠR + .
Az ( 1)
Bz ( 2)
i.e AC.BD = AB.CD + AD. BC
fi |z 1
– z 3
| ◊ |z 2
– z 4
|
= |z 1
– z 2
| ◊ |z 3
– z 4
| + |z 1
– z 4
| ◊ |z 2
– z 4
|.
EXERCISES
LEVEL I
ABC OF COMPLEX NUMBERS
(Problems based on Fundamentals)
1. Find the value of i n + i n + 1 + i n + 2 + i n + 3 , n ΠI.
2. Find the value of i 2010 + i 2011 + i 2012 + i 2013 .
n
Ê1
+ iˆ 3. Find the smallest integer n for which Á = 1
Ë1
- i
˜ .
2013
¯
1
4. Find the sum of ( n n
 i + i + ).
n=
1
5. Find the value of i P + i Q + i R + i S where P, Q, R, S are
four consecutive integers.
6. Find the value of i 2015 + i 2016 + i 2017 + i 2018 .
7. If
2016 2018
k
p
 i +  i = x+ iy, i = -1, find x + y + 2.
k= 0 p=
0
8. Find the smallest positive integer n for which (1 + i) 2n =
(1 – i) 2n .
9. Find the value of (1 + i) 5 + (1 + i 3 ) 5 + (1 + i 5 ) 7 + (1 + i 7 ) 7 .
10. Let z = (n + i) 4 . Find the number of integral values of n
for which z is an integer.
11. If z = 1 + i, find the multiplicative inverse of z 2 .
1+
2i
12. If z = , find the multiplicative inverse of z.
3 - 4i