Manipal Entrance Test 2014 Math Paper

Let Q be the image of the point P(

^

i

+3

^

k

) in the plane r.(

^

i

+

^

j

+

^

k

)=1 Then PQ is normal to the since PQ passes through P and is normal to the given plane, therefore equation of PQ is r=(

^

i

+3

^

k

)+λ(

^

i

+

^

j

+

^

k

)
since, Q lies on the line PQ, so, let the position vector of Q ‌be(

^

i

+3

^

k

)+λ(

^

i

+

^

j

+

^

k

)
⇒(1+λ)

^

i

+λ

^

j

+(3+λ)

^

k

Since Ft is the mid point of PQ, therefore position vector of R is
=

(1+λ) ^ i +λ ^ j +(3+λ) ^ k + ^ i +3 ^ k

2

or (

λ+2

2

)

^

i

+ (

λ

2

)

^

j

+ (3+

λ

2

)

^

k

or (

λ

2

+1 )

^

i

+ (

λ

2

)

^

j

+ (3+

λ

2

)

^

k

since, R less on the plane r.(

^

i

+

^

j

+

^

k

)=1
Therefore,
[ (

λ

2

+1 )

^

i

+ (

λ

2

)

^

j

+ (3+

λ

2

)

^

k

] .(

^

i

+

^

j

+

^

k

)=1
⇒ ( λ2+1+

λ

2

+3+

λ

2

)=1
⇒ λ=−2
So. the postion vector of Q is
(

^

i

+3

^

k

)−2(

^

i

+

^

j

+

^

k

) =

^

i

−2

^

j

+

^

k