If the diagonals of the quadrilateral formed by lines `l_1x+m_1y+n_1=0,` `l_2x+m_2y+n_2=0,` `l_1x+m_1y+n_1'=0,` and `l_2x+m_2y+n_2'=0,` are perpendicular, then value of `l_1^2-l_2^2+m_1^2-m_2^2.`

Show that the diagonals of the parallelogram whose sides are l x+m y+n=0,l x+m y+n^(prime)=0,m x+l y+n=0 and m x+l y+n^(prime)=0 include an angle pi//2 .

The area (in square units ) of the quadrilateral formed by two pairs of lines l^(2) x^(2) - m^(2) y^(2) - n (lx + my) = 0 and l^(2) x^(2)- m^(2) y^(2) + n (lx - my ) = 0 , is

The area (in square units ) of the quadrilateral formed by two pairs of lines l^(2) x^(2) - m^(2) y^(2) - n (lx + my) = 0 and l^(2) x^(2)- m^(2) y^(2) + n (lx - my ) = 0 , is

If the direction cosines of two lines are l_(1), m_(1), n_(1) and l_(2), m_(2), n_(2) , then find the direction cosine of a line perpendicular to these lines.

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 are proportional to l_1+l_2,m_1+m_2, n_1+n_2dot Statement 2: The angle between the two intersection lines having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 is given by costheta=l_1l_2+m_1m_2+n_1n_2dot

The line l_1 x+ m_1 y+n=0 and l_2 x + m_2 y+n=0 will cut the coordinate axes at concyclic points if (A) l_1 m_1 = l_2 m_2 (B) l_1 m_2 = l_2 m_1 (C) l_1 l_2 = m_1 m_2 (D) l_1 l_2 m_1 m_2 = n^2

The direction ratios of the bisector of the angle between the lines whose direction cosines are l_1,m_1,n_1 and l_2,m_2,n_2 are (A) l_1+l_2,m_1+m_2+n_1+n_2 (B) l_1-l_2,m_1-m_2-n_1-n_2 (C) l_1m_2-l_2m_1,m_1n_2-m_2n_1,n_1l_2-n_2l_1 (D) l_1m_2+l_2m_1,m_1n_2+m_2n_1,n_1l_2+n_2l_1

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

If the direction cosines of two lines are (l_(1), m_(1), n_(1)) and (l_(2), m_(2), n_(2)) and the angle between them is theta then l_(1)^(2)+m_(1)^(2)+n_(1)^(2)=1=l_(2)^(2)+m_(2)^(2)+n_(2)^(2) and costheta = l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2) If the angle between the lines is 60^(@) then the value of l_(1)(l_(1)+l_(2))+m_(1)(m_(1)+m_(2))+n_(1)(n_(1)+n_(2)) is

If l_1^2+m_1^2+n_1^2=1 etc., and l_1 l_2+m_1 m_2+n_1 n_2 = 0 , etc. and Delta=|(l_1,m_1,n_1),(l_2,m_2,n_2),(l_3,m_3,n_3)| then