Prove that the lines `x = py + q; z = ry + s and x = p'y + q'; z = r'y + s'` are perpendicular If `p p' + q q' + 1 = 0`

Prove that the lines x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular if pp' + rr' + 1 = 0.

Find the condition that the lines x=py +q, z=ry +s and x=p'y +q', z= r'y+s' may be perpendicular to each other.

Prove that the lines x=py+q, z=ry+s and x=p'y+q'z=r'y+s' are at right angles if pp'+rr'+1=0.

Prove that the value of determinants is zero: |a-b b-c c-a x-y y-z z-x p-q q-r r-p|

The locus of the orthocentre of the triangle formed by the lines : (1 + p)x - py + p(1+ p) = 0, (1 + q)x- qy + q(1+ q) = 0, and y = 0 , where p ne q , is :

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Let image of the line (x-1)/3=(y-3)/5=(z-4)/2 in the plane 2x-y+z+3=0 be Ldot A plane 7x+p y+q z+r=0 is such that it contains the line L and perpendicular to the plane 2x-y+z+3=0 then p=1 (b) q=42 p+q+r=30 (d) p+q+r=0