The lengths of the perpendiculars from the points `(m^(2), 2m), (mn, m+n)` and `(n^(2), 2n)` to the line `x+sqrt3y+3=0` are in

Options

A. Arithmetic progression
B. Geometric progression
C. Harmonic progression
D. None of these

Answer

To solve the problem, we need to find the lengths of the perpendiculars from the points \( (m^2, 2m) \), \( (mn, m+n) \), and \( (n^2, 2n) \) to the line \( x + \sqrt{3}y + 3 = 0 \) and determine if these lengths are in Arithmetic Progression (AP), Geometric Progression (GP), Harmonic Progression (HP), or none.

### Step-by-Step Solution:

1. **Identify the line equation**: The given line is \( x + \sqrt{3}y + 3 = 0 \).

2. **Formula for the length of the perpendicular**: The length \( P \) of the perpendicular from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is given by:
\[ P = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
For our line, \( a = 1 \), \( b = \sqrt{3} \), and \( c = 3 \).

3. **Calculate \( P_1 \)**: For the point \( (m^2, 2m) \):
\[ P_1 = \frac{|1 \cdot m^2 + \sqrt{3} \cdot 2m + 3|}{\sqrt{1^2 + (\sqrt{3})^2}} = \frac{|m^2 + 2\sqrt{3}m + 3|}{2} \]
This can be rewritten as:
\[ P_1 = \frac{|(m + \sqrt{3})^2|}{2} \]

4. **Calculate \( P_2 \)**: For the point \( (mn, m+n) \):
\[ P_2 = \frac{|1 \cdot mn + \sqrt{3} \cdot (m+n) + 3|}{2} = \frac{|mn + \sqrt{3}(m+n) + 3|}{2} \]

5. **Calculate \( P_3 \)**: For the point \( (n^2, 2n) \):
\[ P_3 = \frac{|1 \cdot n^2 + \sqrt{3} \cdot 2n + 3|}{2} = \frac{|n^2 + 2\sqrt{3}n + 3|}{2} \]
This can be rewritten as:
\[ P_3 = \frac{|(n + \sqrt{3})^2|}{2} \]

6. **Check the relationship between \( P_1, P_2, P_3 \)**: We need to check if \( P_1, P_2, P_3 \) are in GP. For them to be in GP, the following condition must hold:
\[ P_1 \cdot P_3 = P_2^2 \]

Substituting the expressions we found:
\[ \frac{|(m + \sqrt{3})^2|}{2} \cdot \frac{|(n + \sqrt{3})^2|}{2} = \left(\frac{|mn + \sqrt{3}(m+n) + 3|}{2}\right)^2 \]

Simplifying gives:
\[ |(m + \sqrt{3})(n + \sqrt{3})| = |mn + \sqrt{3}(m+n) + 3| \]

7. **Conclusion**: Since we have shown that \( P_1 \cdot P_3 = P_2^2 \), we conclude that \( P_1, P_2, P_3 \) are in Geometric Progression (GP).

### Final Answer:
The lengths of the perpendiculars from the points to the line are in **Geometric Progression (GP)**.