Distance formula using Complex Numbers

Q.

Area of the triangle formed by the lines $y^2-9xy+18x^2=0$ and y=9 is

1. 

2. 

3. 

4. 

Q.

Show that the centroid of the triangle with vertices A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is
((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3).

Q. The number of points with integral coordinates that are interior to the circle $x^2+y^2=16$ is

1. 43
2. 49
3. 45
4. 51

Q.

If the vertices of triangle are (0, 2), (1, 0) and (3, 1), then the triangle is

1. Equilateral

2. Isosceles

3. Right angled

4. Isosceles right angled

Q.

The points $(\frac{a}{\sqrt{3}},a),(\frac{2a}{\sqrt{3}},2a),(\frac{a}{\sqrt{3}},3a)$ are the vertices of

1. An equilateral triangle

2. An isosceles triangle

3. A right angled triangle

4. None of these

Q. If $|z_1|=12$ and $|z_2-3-4i|=5$, then the minimum value of $|z_1-z_2|$ is .

1. \N
2. 2
3. 5

Q. The following table shows the signs of coordinates in eight octants.

$\begin{array}{ccccccccc}& I& II& III& IV& V& VI& VII& VIII\\ x& +& -& -& +& +& -& -& +\\ y& +& +& -& -& +& +& -& -\\ z& +& +& +& +& -& -& -& -\end{array}$

In which plane does the point (0, 5, -4) lie?

Q. For all complex numbers $z_1,z_2$ satisfying $|z_1|=12and|z_2-3-4i|$ = 5, the minimum value of $|z_1-z_2|$ is
[IIT Screening 2002]

1. 17

2. 0

3. 2

4. 7

Q. The following table shows the signs of coordinates in eight octants.

$\begin{array}{ccccccccc}& I& II& III& IV& V& VI& VII& VIII\\ x& +& -& -& +& +& -& -& +\\ y& +& +& -& -& +& +& -& -\\ z& +& +& +& +& -& -& -& -\end{array}$

If a point lies in xy-plane then what is its z-coordinate?

Q.

The points $(\frac{a}{\sqrt{3}},a),(\frac{2a}{\sqrt{3}},2a),(\frac{a}{\sqrt{3}},3a)$ are the vertices of

1. An equilateral triangle

2. An isosceles triangle

3. A right angled triangle

4. None of these

Q. If A, B, C are represented by 3 + 4i, 5 - 2i , -1 + 16i, then A, B, C are
[RPET 1986]

1. Collinear

2. Vertices of equilateral triangle

3. Vertices of isosceles triangle

4. Vertices of right angled triangle

Q. Length of the line segment joining the points -1 -i and 2 + 3i is

1. – 5

2. 15

3. 5

4. 25

Q.

Let the complex numbers z₁, z₂ and z₃ be the vertices of an equilateral triangle. Let z₀ be the circumcentre of the triangle, then $z_1^2+z_2^2+z_3^2$=

1. 

2. 

3. 

4.