UPSC NDA Math Model Paper 1

Here, we have to find the equation of the plane passing through the intersection of the planes

→

r

.(

^

i

+3

^

j

−

^

i

)−0 and

→

r

.(

^

j

+2

^

k

)−0 and passing through the point (2, 1, -1).
As we know that, the equation of the plane through the intersection of two planes

→

r

.n1=q1 and

→

r

.n2=q2 is given by

→

r

.(

→

n

1+λ

→

n

2)=q1+λq2
Here,

→

n

1=

^

i

+3

^

j

−

^

k

,

→

n2

=

^

j

+2

^

k

,q1=0 and q2=0
So, the equation of the required plane is:

→

r

.[

^

i

+(3+λ)

^

j

+(−1+2λ)

^

k

]=0 ..........(1)
∵ The plane represented by (1) passes through the point (2,1,-1)
So,

→

r

=2

^

i

+

^

j

−

^

k

will satisfy the equation (1)
⇒(2

^

i

+

^

j

−

^

k

).(

^

i

+(3+λ)

^

j

+(−1+2λ)

^

k

)=0
⇒2+3+λ+1−2λ=0
⇒λ=6
By substituting λ=6 in equation (1) we get,

→

r

.(

^

i

+9

^

j

+11

^

k

)=0
Hence, the equation of the required plane is

→

r

.(

^

i

+9

^

j

+11

^

k

)=0