If `z =(lambda+3)+isqrt((5-lambda^2))` ; then the locus of z is a) a straight line b) a semicircle c) an ellipse d) a parabola

Options

A. ellispe
B. semicircle
C. parabola
D. none of these

Explanation

To find the locus of the complex number \( z = (\lambda + 3) + i \sqrt{5 - \lambda^2} \), we can follow these steps:

### Step 1: Express \( z \) in terms of \( x \) and \( y \)

We start by writing \( z \) in the form \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. From the given equation:

\[ z = (\lambda + 3) + i \sqrt{5 - \lambda^2} \]

We can identify:

\[ x = \lambda + 3 \]
\[ y = \sqrt{5 - \lambda^2} \]

### Step 2: Express \( \lambda \) in terms of \( x \)

From the equation for \( x \):

\[ \lambda = x - 3 \]

### Step 3: Substitute \( \lambda \) into the equation for \( y \)

Now substitute \( \lambda \) into the equation for \( y \):

\[ y = \sqrt{5 - (x - 3)^2} \]

### Step 4: Square both sides to eliminate the square root

To eliminate the square root, we square both sides:

\[ y^2 = 5 - (x - 3)^2 \]

### Step 5: Rearrange the equation

Rearranging gives:

\[ y^2 + (x - 3)^2 = 5 \]

### Step 6: Identify the locus

The equation \( y^2 + (x - 3)^2 = 5 \) represents a circle with center at \( (3, 0) \) and radius \( \sqrt{5} \). Since \( y \) must be non-negative (as \( y = \sqrt{5 - \lambda^2} \)), the locus is actually the upper half of this circle, which is a semicircle.

### Conclusion

Thus, the locus of \( z \) is a semicircle.

### Final Answer: b) a semicircle

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