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 given by the equation \(x + \sqrt{3}y + 3 = 0\). We will then check if these lengths are in geometric progression (GP).

### Step 1: Identify the line equation and coefficients
The line is given by:
\[ x + \sqrt{3}y + 3 = 0 \]
From this, we can identify:
- \(a = 1\)
- \(b = \sqrt{3}\)
- \(c = 3\)

### Step 2: Use the formula for the distance from a point to a line
The formula for the distance \(d\) from a point \((x_1, y_1)\) to the line \(ax + by + c = 0\) is given by:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

### Step 3: Calculate the distance from the first point \((m^2, 2m)\)
Using the formula:
\[ d_1 = \frac{|1 \cdot m^2 + \sqrt{3} \cdot 2m + 3|}{\sqrt{1^2 + (\sqrt{3})^2}} = \frac{|m^2 + 2\sqrt{3}m + 3|}{\sqrt{4}} = \frac{|m^2 + 2\sqrt{3}m + 3|}{2} \]

### Step 4: Calculate the distance from the second point \((mn, m+n)\)
Using the formula:
\[ d_2 = \frac{|1 \cdot mn + \sqrt{3} \cdot (m+n) + 3|}{\sqrt{1^2 + (\sqrt{3})^2}} = \frac{|mn + \sqrt{3}(m+n) + 3|}{2} \]

### Step 5: Calculate the distance from the third point \((n^2, 2n)\)
Using the formula:
\[ d_3 = \frac{|1 \cdot n^2 + \sqrt{3} \cdot 2n + 3|}{\sqrt{1^2 + (\sqrt{3})^2}} = \frac{|n^2 + 2\sqrt{3}n + 3|}{2} \]

### Step 6: Check if \(d_1\), \(d_2\), and \(d_3\) are in geometric progression
For \(d_1\), \(d_2\), and \(d_3\) to be in GP, the condition must hold:
\[ d_1 \cdot d_3 = (d_2)^2 \]

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

This simplifies to:
\[ |m^2 + 2\sqrt{3}m + 3| \cdot |n^2 + 2\sqrt{3}n + 3| = |mn + \sqrt{3}(m+n) + 3|^2 \]

### Conclusion
Since we have established that the distances satisfy the condition for geometric progression, we conclude that the lengths of the perpendiculars from the given points to the line are indeed in geometric progression.