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Prove that the lengths of the perpendicu...

Prove that the lengths of the perpendiculars from the points `(m^2,2m),(m m^(prime),m+m^(prime)),` and `(m^('2),2m^(prime))` to the line `x+y+1=0` are in GP.

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To prove that the lengths of the perpendiculars from the points \( (m^2, 2m), (m m', m + m'), (m'^2, 2m') \) to the line \( x + y + 1 = 0 \) are in geometric progression (GP), we will follow these steps: ### Step 1: Identify the formula for the length of the perpendicular from a point to a line The formula for the length of the perpendicular \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] In our case, the line is \( x + y + 1 = 0 \), so \( A = 1, B = 1, C = 1 \). ...
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