##
**
**

**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
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
**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
Solution

**Q4. **The vector equation of the plane passing through the origin and the line of intersection of the plane r ⃗∙a ⃗=Î» and r ⃗∙b ⃗=Î¼ is
**Q6. **If the direction cosines of a line are (1/c,1/c,1/c), then
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)
Solution

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

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

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

**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

Solution

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

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

**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

Solution

**Q4.**The vector equation of the plane passing through the origin and the line of intersection of the plane r ⃗∙a ⃗=Î» and r ⃗∙b ⃗=Î¼ is

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

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

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)

Solution

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

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

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

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.

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

Solution

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

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