3-D COORDINATE GEOMETRY QUIZ-8

Q1. If projection of a line on x,y and z-axes are 6,2 and 3 respectively, then direction cosines of the line is

•  (6/11,2/11.5/11)
•  (6/11,2/11.3/11)
•  (7/6,7/2,7/3)
•  (6/7,2/7,3/7)

Solution
Direction cosines =(6/√(36+4+9),2/√(36+4+9),3/√(36+4+9)) =(6/7,2/7,3/7)

Q2.The shortest distance between the lines (x-3)/3=(y-8)/(-1)=(z-3)/1 and (x+3)/(-3)=(y+7)/2=(z-6)/4 is

•  5√30
•  3√30
•  2√30
•  √30

Solution

 

Q3.  The equation of the plane through the line of intersection of planes ax+by+cz+d=0,a^'x+b^'y+c^'z+d^'=0 and parallel to the line y=0,z=0 is

•   (ab^'-a'b)x+(bc^'-b'c)y+(ad^'-a'd)=0
•  (ab^'-a'b)x+(bc^'-b'c)y+(ad^'-a'd)z=0
•  None of these
•  (ab^'-a'b)y+(ac^'-a'c)z+(ad^'-a'd)=0

Solution

Q4. The vector equation of the plane passing through the origin and the line of intersection of the plane r ⃗∙a ⃗=λ and r ⃗∙b ⃗=μ is

•  r ⃗∙(λ a ⃗-μ b ⃗ )=0
•  r ⃗∙(λ a ⃗+μ b ⃗ )=0
•  r ⃗∙(λ b ⃗+μ a ⃗ )=0
•  r ⃗∙(λ b ⃗-μ a ⃗ )=0

Solution

Q5.The image (or reflection) of the point (1,2,-1) in the plane r ⃗∙(3i ̂-5j ̂+4k ̂ )=5 is

•  (73/25,-6/5,39/25)
•  (73/25,6/5,39/25)
•  (-1,-2,1)
•  None of these

Solution
 

 

Q6. If the direction cosines of a line are (1/c,1/c,1/c), then

•  c=±√2
•  c=±√3
• c>2
•  0

Solution
Since,DC^' sof a line are(1/c,1/c,1/c) ∴ (1/c)^2+(1/c)^2+(1/c)^2=1 ⟹ c^2=3⟹c=± √3

Q7.The xy-plane divides the line joining the points (-1,3,4)and (2,-5,6)

•  Internally in the ratio 2 : 3
•  Internally in the ratio 3 : 2
•  Externally in the ratio 2 : 3
•  Externally in the ratio 3 : 2

Solution
Suppose xy-plane divides at the line joining the given points in the ratio λ : 1 . The coordinate of the points of division are[(2λ-1)/(λ+1),(-5λ+3)/(λ+1),(6λ+4)/(λ+1)] Since the point lies on the xy-plane ∴(6λ+4)/(λ+1)=0⟹λ=(-2)/3

Q8.The equation of the plane containing the line (x-x1)/l=(y-y1)/m=(z-z1)/n is

•  ax1+by1+cz1=0
•  al+bm+cn=0
•  a/l=b/m=c/n
•  lx1+my1+nz1=0

Solution
Required plane contains the given line, so normal to the plane must be perpendicular to the line and the condition for the same is al+bm+cn=0.

Q9.If x coordinate of a point P of line joining the points Q(2,2,1) and R(5,2,-2) is 4, then the z coordinate of P is

•  -2
•  2
•  1
•  -1

Solution
Suppose P divides QR in the ratio λ:1. Then, coordinates of P are ((5λ+2)/(λ+1),(2λ+2)/(λ+1),(-2λ+1)/(λ+1)) Since, the x coordinates of P is 4 ie,(5λ+2)/(λ+1)=4 ⇒ λ=2 So, z coordinate of P is (-2λ+1)/(λ+1)=(-4+1)/(2+1)=-1

Q10. The equation of the plane through the points (1, 2, 3), (-1,4,2)and (3,1,1) is

•  5x+y+12z-23=0
•  5x+6y+2z-23=0
•  x+6y+2z-13=0
• x+y+z-13=0

Solution