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 solve the problem of finding 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 break it down step by step.

### Step 1: Rewrite the first equation
The first equation is given by:
\[ \arg(z - 2 - 7i) = \cot^{-1}(2) \]
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite the equation as:
\[ \arg((x - 2) + i(y - 7)) = \cot^{-1}(2) \]
The argument can be expressed as:
\[ \frac{y - 7}{x - 2} = \tan(\cot^{-1}(2)) = \frac{1}{2} \]
This leads to the equation:
\[ 2(y - 7) = x - 2 \]
Simplifying this gives:
\[ x = 2y - 12 \quad \text{(Equation 1)} \]

### Step 2: Rewrite the second equation
The second equation is:
\[ \arg\left(\frac{z - 5i}{z + 2 - i}\right) = \pm \frac{\pi}{2} \]
This means that the fraction is purely imaginary. Therefore, we have:
\[ \frac{z - 5i}{z + 2 - i} = i \cdot k \quad \text{for some real } k \]
This implies:
\[ z - 5i = i k (z + 2 - i) \]
Expanding this gives:
\[ z - 5i = ikz + 2ik - k \]
Rearranging terms leads to:
\[ z - ikz = 5i + 2ik - k \]
Factoring out \( z \):
\[ z(1 - ik) = 5i + 2ik - k \]
Thus, we can express \( z \) as:
\[ z = \frac{5i + 2ik - k}{1 - ik} \]
For \( k = 0 \), we find:
\[ z = 5i \quad \text{(Equation 2)} \]
For \( k \neq 0 \), we can analyze the imaginary part to find the conditions.

### Step 3: Find points of intersection
Now we have two equations:
1. \( x = 2y - 12 \) (from Equation 1)
2. \( z = 5i \) (from Equation 2)

Substituting \( y = 5 \) into Equation 1:
\[ x = 2(5) - 12 = 10 - 12 = -2 \]
Thus, we have the point \( z = -2 + 5i \).

### Step 4: Conclusion
The curves intersect at the point \( z = -2 + 5i \). Since we have found only one point of intersection, the number of points of intersection is:
\[ \text{Number of points of intersection} = 1 \]