Let a line passing through the point `(-1,2,3)` intersect the lines `L_(1):(x-1)/(3)=(y-2)/(2)=(z+1)/(-2)` at `M(alpha,beta,gamma)` and `L_(2):(x+2)/(-3)=(y-2)/(-2)=(z-1)/(4)` at `N(a,b,c)` Then,the value of `((alpha+beta+gamma)^(2))/((a+b+c)^(2))` equals__________

Let (alpha,beta,gamma) be the mirror image of the point (2,3,5) in the line (x-1)/(2)=(y-2)/(3)=(z-3)/(4) .Then, 2 alpha+3 beta+4 gamma is equal to

If the lines (x-3)/(2)=(y-5)/(2)=(z-4)/(lambda) and (x-2)/(lambda)=(y-6)/(4)=(z-5)/(2) intersect at a point (alpha, beta, gamma) , then the greatest value of lambda is equal to

The foot of perpendicular from (1,2,6) to line (x-3)/(2)=(y+1)/(2)=(z-1)/(-1) is the point (alpha,beta,gamma) then (alpha+beta+gamma)=

Let the line (x-2)/(3)=(y-1)/(-5)=(z+2)/(2) lie in the plane x+3y-alpha z+ beta=0 then the value of alpha+beta=

(i) Find the equations of the straight line passing through the point (2,3,-1) and is perpendicular to the lines : ( x-2)/(2) = (y + 1)/(1) = (z - 3)/(-3) and (x - 3)/(1) = (y + 2)/(1) = (z - 1)/(1) . (ii) Find the equation of the line which intersects the lines : (x + 2)/(1) = (y - 3)/(2) = (z + 1)/(4) and (x - 1)/(2) = (y - 2)/(3) = (z - 3)/(4) Perpendicular and passes through the point (1,1,1) .

If the mirror image of the point P(3, 4, 9) in the line (x-1)/3=(y+1)/2=(z-2)/1 is (alpha, beta) then the value of 14(alpha+beta+gamma) is.

A line L passing through (1, 2, 3) and perpendicular to the line L_(1):(x-1)/(-2)=(y+1)/(3)=(z-5)/(4) is also intersecting the line L_(1) . If the line L intersects the plane 2x+y+z+6=0 at point (alpha, beta, gamma) , then the value of 2020alpha+beta+2gamma is equal to

If for some alpha in R, the lines L_1 : (x + 1)/(2) = (y-2)/(-1) = ( z -1)/(1) and L_2 : (x + 2)/(alpha) = (y +1)/(5 - alpha) = (z + 1)/(1) are coplanar , then the line L_2 passes through the point :

If image of point A(2,3,1) in the lines (x-1)/2=(y-4)/1=(z+3)/-1 lies on the plane alphax+betay+gammaz=24 also the line (x-1)/1=(1-y)/2=(z-6)/15 lies in the plane then alpha+beta+gamma= ?

Two lines L_(1):x=5,(y)/(3-alpha)=(z)/(-2) and L_(2):x=alpha,(y)/(-1)=(z)/(2-alpha) are coplanar.Then alpha can take value (s) a.1 b.2 c.3 d.4