If `p_(1),p_(2),p_(3)` be the length of perpendiculars from the points `(m^(2),2m),(mm',m+m')` and `(m^('2),2m')` respectively on the line `xcosalpha+ysinalpha+(sin^(2)alpha)/(cosalpha)=0` then `p_(1),p_(2),p_(3)` are in:

Answer

To solve the problem, we need to find the lengths of the perpendiculars \( p_1, p_2, p_3 \) from the given points to the specified line. Let's go through the steps systematically.

### Step 1: Identify the points and the line equation
The points are:
- \( A(m^2, 2m) \)
- \( B(mm', m + m') \)
- \( C(m'^2, 2m') \)

The line equation is given by:
\[ x \cos \alpha + y \sin \alpha + \frac{\sin^2 \alpha}{\cos \alpha} = 0 \]

### Step 2: Use the formula for the length of the perpendicular from a point to a line
The formula for the perpendicular distance \( p \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is:
\[ p = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

In our case, \( A = \cos \alpha \), \( B = \sin \alpha \), and \( C = \frac{\sin^2 \alpha}{\cos \alpha} \).

### Step 3: Calculate \( p_1 \)
For point \( A(m^2, 2m) \):
\[ p_1 = \frac{|\cos \alpha \cdot m^2 + \sin \alpha \cdot 2m + \frac{\sin^2 \alpha}{\cos \alpha}|}{\sqrt{\cos^2 \alpha + \sin^2 \alpha}} \]
Since \( \sqrt{\cos^2 \alpha + \sin^2 \alpha} = 1 \):
\[ p_1 = |\cos \alpha \cdot m^2 + 2m \sin \alpha + \frac{\sin^2 \alpha}{\cos \alpha}| \]

### Step 4: Calculate \( p_2 \)
For point \( B(mm', m + m') \):
\[ p_2 = \frac{|\cos \alpha \cdot (mm') + \sin \alpha \cdot (m + m') + \frac{\sin^2 \alpha}{\cos \alpha}|}{\sqrt{\cos^2 \alpha + \sin^2 \alpha}} \]
Thus,
\[ p_2 = |\cos \alpha \cdot (mm') + \sin \alpha \cdot (m + m') + \frac{\sin^2 \alpha}{\cos \alpha}| \]

### Step 5: Calculate \( p_3 \)
For point \( C(m'^2, 2m') \):
\[ p_3 = \frac{|\cos \alpha \cdot m'^2 + \sin \alpha \cdot 2m' + \frac{\sin^2 \alpha}{\cos \alpha}|}{\sqrt{\cos^2 \alpha + \sin^2 \alpha}} \]
Thus,
\[ p_3 = |\cos \alpha \cdot m'^2 + 2m' \sin \alpha + \frac{\sin^2 \alpha}{\cos \alpha}| \]

### Step 6: Analyze the relationship between \( p_1, p_2, p_3 \)
To determine if \( p_1, p_2, p_3 \) are in a specific progression, we can check if:
\[ p_2^2 = p_1 \cdot p_3 \]
This indicates that \( p_1, p_2, p_3 \) are in geometric progression (GP).

### Conclusion
Thus, the lengths of the perpendiculars \( p_1, p_2, p_3 \) are in geometric progression.