x+ y - z+1= 0, 4x+ y - 2z+2 = 0 are (±1/√14, ±2/√14, ±3/√14).
2) Show that the equation of the straight line through the point (3,4,5) which is equally inclined to the axes are x- 3= y -4= z -5..
3) a) Find the equations of the straight lines passing through the points
i) (3,-9,4) and (-9,5,-4). (x+9)/6= (y -5)/-7= (z +4)/4
ii) (-7,5,3) and (2,6,8). (x+7)/9 = (y -5)/1 = (z -8)/5
b) Show that the equation of the median AD of the triangle whose vertices are A(3,4,8), B(1,-6,2), C(1,4,-2) are
(x-3)/2 = (y -4)/5 = (z -7)/8
4a) Show that the three points (-1,5,3),(5,1,5) and (8,-1,6) are collinear.
b) Show that the straight line through the points (a,b,c) and (a', b', c') passes through the origin, if aa' + bb' + cc' = pp' where p and p' are the distances of the points from the origin.
5) Put the equations of the straight lines in symmetrical form as given by
a) x + 5y - z -7= 0, 2x - 5y + 3z +1= 0. (x-2)/2 = (y -1)/-1 = z/-3
b) x + y + z +1 = 0, 4x + y - 2z + 2= 0. (x+1/3)/1 = (y+ 2/3)/-2 = z/1
6) a) Find the image of the point
i) (1,-2,3) in the plane 2x - 3y + 2z +3= 0. (-3,4,-1)
ii) (1,3,4) in the plane 2x - y + z +3= 0. (-3,5,2)
b) Find the image of the straight line
(x -1)/3 = (y -3)/5 = (z -4)/2 in the plane 2x - y + z+3= 0
7) a) Find the equations of the straight line through the point (α, β, γ) which is
i) parallel to the z-axis. (x - α)/0 = (y- β)/0 = (z -γ)/1
ii) Perpendicular to the z-axis. (x - α)/l = (y- β)/m = (z -γ)/0
b) Find the equations of the straight line passing through the point (1,2,3) and parallel to the straight line.
i) x/2= y/4= z/3. (x - 1)/2 = (y- 2)/4 = (z -3)/3
ii) x - y + 2z - 5= 0= 3x + y + z = 6. (x - 1)/-3 = (y- 2)/5 = (z -3)/4
c) Find the equations of the straight line through the point (1,2,3) and parallel to the straight line joining the points (-4,7,2) and (5,-3,-2). (x - 1)/-9 = (y- 2)/10 = (z -3)/4
d) Find the equations of the straight line through the point (8,9,10) and perpendicular to each of the straight lines.
(x - 2)/3 = (y- 3)/2 = (z +4)/4 and (x + 1)/5 = (y- 2)/-6 = (z +3)/2. (x - 8)/2 = (y- 9)/1 = (z +10)/-2
8) Show that the equations of the straight line passing through the point (1,-2,3) and perpendicular to the plane 2x + y + 3z= 4 is (x - 1)/2 = (y + 2)/1 = (z -3)/3.
9) Show that the straight line (x - 1)/-1 = (y+ 4)/3 = (z +5)/2 meets the plane 2x - 3y + 4z = 0 at the point (4/3,-5,-17/3)
10) a) Show that the distance of the point of intersection of the straight line (x - 2)/3 = (y +1)/4 = (z -1)/2 and the plane x+ y +z= 12 from the point (-1,5,10) is 2√19 units.
b) Find the coordinates of the point in which the straight line (x - 1)/2 = (y +1)/-1 = (z )/3 intersects the plane 3x + 2y - z= 5. (9,-5,12)
c) Find the point where the straight line through the points (5,-2,3) and (3,0,1) pieces the xy-plane. (2,1,0)
d) Find the coordinates of the point where the straight line x+ 3y - z= 6, y- z= 4 meets the plane 2x + 2y +z= 0. (2,0,-4)
e) Find the points where the straight line (x - a)/l = (y- b)/m = (z -c)/n meets the coordinates plane. (0,b - am/l, c- an/l), (a- bl/m, 0, c- bn/m), (a- cl/n, b - cm/n, 0)
f) A straight line is drawn through the points (-6,6,-5) and (12,-6,1). Find the points in which it meets the coordinate planes. (9,-4,0),(3,0,-2),(0,2,-3)
11a) Show that the distance of the point (3,-4,5) from the plane 2x + 5y - 6z= 0 measured along the straight line whose direction ratios are+2,-1,-2) is 12 units.
b) Find the distance of the point (1,-2,3) from the plane x- y + z= 5 measured parallel to the straight line x/2= y/3 =z/6. 7/5 units
c) Show that the distance of the point (3,8,2) from the straight line (x - 1)/2 = (y- 3)/4 = (z -2)/3 measured parallel to the plane 3x + 2y - 2z+ 15= 0 is 7 units.
12) a) Show that the foot of the Perpendicular from the point (-1,3,2) to the plane x+ 2y + 2z -3=0 is (-5/3,5/3,2/3)
b) Show that the equations of the projection of the straight line (x - 1)/2 = (y- 2)/-1 = (z -3)/4 on the plane x+ 2y + z= 6 are (x - 3)/4 = (y +2)/-7 = (z -7)/10.
13) a) Are the two straight lines (x - 2)/3 = (y- 3)/2 = (z +4)/4 and (x + 1)/5 = (y- 2)/-6 = (z +3)/2 perpendicular to each other? No
b) Find whether the following straight lines are mutually perpendicular
(x - 1)/3 = (y+ 2)/2 = (z -6)/5 and 2x + y - 3z -2=0= 3x + 2y + 5z +7. Yes
c) Show that the straight lines x= at+ b, z= cy+ d and x= a'y + b', z= c'y + d' are at right angles, if aa' + cc' +1= 0.
14) Show that the angle between the straight line (x - 4)/7 = (y- 1)/4 = (z +3)/4 and the plane x - 2y - 2z= 8 is sin⁻¹(1/3).
15) a) Show that the straight lines 3x + 2y - 3z +5= 0= x - 2y +z -3 and 12x - 4y - 3z +13=0 9x + 2y - 6z +3 are parallel.
b) Show that the straight line 2x + 3y - z+3=0= 3x - 2y + 2z -6 is not parallel to the z-axis.
16a) Show that the straight line x -1= y-2= (1/2)(z -3) lies on the plane 2x + 4y - 3z = 1.
b) Find the values of b and c for which the straight line (x - 1)/2 = (y- 2)/ -1 = (z +3)/3 lies on the plane 9x + by + cz= 30. 3,-5
17a) Show that the line x - y - z+3=0 = 3x + 3y - z -15 is normal to the plane 2x - y + 3z +4=0.
b) A straight line is given by x+ y+ z= 0, x =y. Show that it is perpendicular to the plane x+ y= 2z.
18) Show that the straight line x= t -2, y= 3- 4t, z= 5t +6 is parallel to the plane x - y - z= 1.
19) a) Show that the straight line 2x + 2y - z -6=0= 2x + 3y - z -8 is parallel to a coordinate plane and find the equation of the plane normal to this straight line and passing through the point where this straight line meets the plane x= 0. x+ 2z+4= 0
b) Show that the equation of the plane through the origin and containing two straight lines whose direction ratios are (1,0,2) and (-1,0,5) is 10x +2y - 5z= 0.
20a) Show that the equation of the plane containing the straight line x+ y+ z-1= 0= 2x + 3y + 4z -5 and perpendicular to the plane x - y + z= 0 is x - z +2=0.
b) Find the equation of the plane which is perpendicular to the plane x+ 2y - z +1= 0 and which contains the line of intersection of the planes x+ 2y + 3z -4=0 and 2x + y + z+2=0. x - 4y - 7z +16=0
21) Show that the equation of the straight line through the point (3,1,-6) and parallel to each of the planes x+ y + 2z -4=0 and 2x - 3y + z +5=0 are (x - 3)/7 = (y- 1)/3 = (z +6)/-5.
b) Obtain the equations of the straight line passing through the point+2,3,5) and parallel to the intersection of the planes x+ 2y -1= 0 and 2y + 3z -5=0. x -2= -2(y -3)= 3(z -5)
c) Find the equations of the straight line through the point (1,2,4) and perpendicular to the straight line 3x + 2y - z-4=0= x - 2y - 2z -5. (x - 1)/-202 = (y- 2)/460 = (z -4)/439
22) a) Show that the plane containing the straight line (x - 1)/3 = (y +6)/4 = (z +1)/2 and parallel to the straight line (x - 1)/2 = (y- 1)/-3 = (z+4)/5 is 26x - 11y - 17z - 109= 0.
b) Show that the equation of the plane through the point+2,3,3) and parallel to the straight lines x -1= 2y -5= 2z and 3x = 4y -11= 3z -4 is x - 4y + 2z +4= 0.
23) a) Show that the plane passing through the point (-2,-2,2) and containing the straight line joining the two points (1,-1,2) and (1,1,1) is x - 3y - 6z +8=0.
b) Show that the equation of the plane through the point (0,7,-7) and containing the straight line (x + 1)/3 = (y- 3)/-2 = (z +2)/-1 is x+ y+ z= 0
c) show that the plane through the point (α, β, γ) and the straight line x= py+ q= rz+ s is given by
x py+ q rz+ s
α pβ + q rγ + s = 0
1 1 1
24) a) Find the equations of the Perpendicular from the point (5,9,3) to the straight line
(x - 1)/2 = (y- 2)/3 = (z -3)/4. Also find the foot of the Perpendicular. (x - 5)/1 = (y- 9)/1 = (z -2)/-2, (3,5,7)
b) Find the equations of the Perpendicular from the point (1,6,3) to the straight line x + y - z + 1= 0= 2x - 7y + 4z-1. Also find the foot of the Perpendicular. (x - 1)/0 = (y- 6)/3 = (z-3)/-2, (1,3,5)
25) a) Find the plane through the point (3,-2,1) and perpendicular to the straight line
2x - 5y - 2z + 6= 0= 4x + y - 17z - 109= 0.
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