JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Three Dimensional Geometry

Let

P

be a point in the first octant, whose image

Q

in the plane

x + y = 3

(that is, the line segment

P Q

is perpendicular to the plane

x + y = 3

and the mid-point of

P Q

lies in the plane

x + y = 3

) lies on the

z

-axis. Let the distance of

P

from the

x

-axis be 5. If

R

is the image of

P

in the

x y

-plane, then the length of

P R

is ____.

Answer: 8

Solution

Let

Q \equiv (0, 0, z_{1})

on the

z

-axis. The image

P

Q

in the plane

x + y = 3

is found using the perpendicular from

Q

\frac{x _{P} - 0}{1} = \frac{y _{P} - 0}{1} = \frac{- 2 ( - 3 )}{2} = 3

\Rightarrow P \equiv (3, 3, z_{1})

Distance from

P

to the

x

-axis is

y_{P}^{2} + z_{1}^{2} = 5

9 + z_{1}^{2} = 5 \Rightarrow z_{1}^{2} = 16 \Rightarrow z_{1} = 4

The image

R

P

in the

x y

-plane is

R \equiv (3, 3, - 4)

.

Length of

P R = 2 z_{1} = 8

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Numerical answer