Prove that a point can be found which is at the same distance from each of the four points :`(a m_1, a/(m_1)), (a m_2, a/(m_2)),(a m_3, a/(m_3))"and"(a/(m_1m_2 m_3), a m_1m_2m_3)`

Prove that the three lines from O with direction cosines l_1, m_1, n_1: l_2, m_2, n_2: l_3, m_3, n_3 are coplanar, if l_1(m_2n_3-n_2m_3)+m_1(n_2l_3-l_2n_3)+n_1(l_2m_3-l_3m_2)=0

In the figure, if m_(1) is at rest, find the relation among m_(1), m_(2) and m_(3) ?

The slope of graph as shown in figure at points 1,2 and 3 is m_(1), m_(2) and m_(3) respectively then

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.

If vec a and vec b are two non-collinear vectors, show that points l_1 vec a+m_1 vec b ,l_2 vec a+m_2 vec b and l_3 vec a+m_3 vec b are collinear if |(l_1,l_2,l_3),(m_1,m_2,m_3),( 1, 1, 1)|=0.

If m+1=(5(m-1))/(3) , then (1)/(m)=

Prove that the lengths of the perpendiculars from the points (m^2,2m),(m m^(prime),m+m^(prime)), and (m^(prime2),2m^(prime)) to the line x+y+1=0 are in GP.

Show that the lines y-m_1 x-c_1 = 0 , y-m_2x-c_2 = 0 and y-m_3 x-c_3 =0 form an isosceles triangle with the first line as base if : (1+m_1 m_2) (m_1 - m_3) + (1+m_1 m_3) (m_1 - m_2) = 0

Solve the linear equations. m - (m- 1)/2 = 1 - (m - 2)/3

If three mutually perpendicular lines have direction cosines (l_1,m_1,n_1),(l_2,m_2,n_2) and (l_3 ,m_3, n_3) , then the line having direction ratio l_1+l_2+l_3,m_1+ m_2+m_3, and n_1 + n_2 + n_3 , make an angle of