The number of points of intersection of the curves represented by `arg(z-2-7i)=cot^(-1)(2)` and arg `((z-5i)/(z+2-i))=pm pi/2`

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To find the number of points of intersection of the curves represented by the equations \( \arg(z - 2 - 7i) = \cot^{-1}(2) \) and \( \arg\left(\frac{z - 5i}{z + 2 - i}\right) = \pm \frac{\pi}{2} \), we will follow these steps:

### Step 1: Convert \( z \) to Cartesian Coordinates
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Analyze the First Curve
The first curve is given by:
\[ \arg(z - 2 - 7i) = \cot^{-1}(2) \]
Substituting \( z \):
\[ \arg((x + iy) - (2 + 7i)) = \cot^{-1}(2) \]
This simplifies to:
\[ \arg((x - 2) + i(y - 7)) = \cot^{-1}(2) \]
The argument can be expressed as:
\[ \frac{y - 7}{x - 2} = \tan(\cot^{-1}(2)) = \frac{1}{2} \]
This gives us the equation of a line:
\[ y - 7 = \frac{1}{2}(x - 2) \]
Rearranging gives:
\[ 2y - 14 = x - 2 \implies x - 2y + 12 = 0 \]
This is our first curve (a straight line).

### Step 3: Analyze the Second Curve
The second curve is:
\[ \arg\left(\frac{z - 5i}{z + 2 - i}\right) = \pm \frac{\pi}{2} \]
This implies that the expression is purely imaginary, meaning:
\[ \frac{z - 5i}{z + 2 - i} \text{ is vertical} \]
This occurs when the real part of the fraction is zero:
\[ \text{Re}(z - 5i) = 0 \implies z - 5i = k(z + 2 - i) \text{ for some real } k \]
This leads to two cases:
1. \( \arg(z - 5i) = \frac{\pi}{2} \) implies \( z = 5i \)
2. \( \arg(z + 2 - i) = -\frac{\pi}{2} \) implies \( z = -2 + i \)

### Step 4: Find the Circle Equation
The second curve can also be interpreted as a circle with diameter endpoints at \( 5i \) and \( -2 + i \). The center of the circle is:
\[ \left(\frac{0 - 2}{2}, \frac{5 + 1}{2}\right) = \left(-1, 3\right) \]
The radius is half the distance between the two points:
\[ \text{Distance} = \sqrt{(-2 - 0)^2 + (1 - 5)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]
Thus, the radius is \( \sqrt{5} \).

### Step 5: Write the Circle Equation
The equation of the circle is:
\[ (x + 1)^2 + (y - 3)^2 = 5 \]

### Step 6: Find Points of Intersection
Now we need to find the intersection of the line \( x - 2y + 12 = 0 \) with the circle \( (x + 1)^2 + (y - 3)^2 = 5 \).

1. Substitute \( y \) from the line equation into the circle equation:
\[ y = \frac{x + 12}{2} \]
2. Substitute into the circle equation:
\[ (x + 1)^2 + \left(\frac{x + 12}{2} - 3\right)^2 = 5 \]
Simplifying this will yield a quadratic equation in \( x \).

### Step 7: Solve the Quadratic Equation
After solving the quadratic equation, we will find the \( x \)-coordinates of the intersection points.

### Step 8: Check Quadrants
Finally, we need to check if the intersection points satisfy the condition that \( \arg(z - 2 - 7i) = \cot^{-1}(2) \) is in the first quadrant.

### Conclusion
After checking the conditions, we find that the total number of points of intersection is **zero**.