Maths · Straight line: Various forms of equations of a line, intersection of lines, angles between two lines

The lengths of the perpendicular from the points ,\left(\boldsymbol{m m}^{\prime

The lengths of the perpendicular from the points \( \left(\boldsymbol{m}^{2}, \boldsymbol{2 m}\right),\left(\boldsymbol{m m}^{\prime}, \boldsymbol{m}+\boldsymbol{m}^{\prime}\right) \) and \( \left(m^{\prime 2}, 2 m^{\prime}\right) \) to the line \( x+y+1=0 \) form \( \cos \alpha+y \sin \alpha+\sin \alpha \tan \alpha=0 \) are in

  • A. an A.P.
  • B. a G.P.
  • C. а н.Р.
  • D. none of these

Step-by-step solution

The perpendicular distances from the three points to the line x+y+1=0 are d1=(m+1)^2/√2, d2=|(m+1)(m'+1)|/√2, d3=(m'+1)^2/√2. These form a geometric progression because d2^2 = d1*d3 for all m,m'.
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