JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Three Dimensional Geometry

Consider a pyramid

O P QR S

located in the first octant

(x \geq 0, y \geq 0, z \geq 0)

with

O

as origin, and

O P

and

O R

along the

x

-axis and the

y

-axis, respectively. The base

O P QR

of the pyramid is a square with

O P = 3

. The point

S

is directly above the mid-point

T

of diagonal

O Q

such that

T S = 3

. Then

(A)the acute angle between $O Q$ and $O S$ is $\frac{π}{3}$
(B)the equation of the plane containing the triangle $O QS$ is $x - y = 0$ ✓ Correct
(C)the length of the perpendicular from $P$ to the plane containing the triangle $O QS$ is $\frac{3}{2}$ ✓ Correct
(D)the perpendicular distance from $O$ to the straight line containing $R S$ is $\frac{15}{2}$ ✓ Correct

Answer: (B), (C), (D)

Solution

The coordinates are:

O (0, 0, 0)

P (3, 0, 0)

Q (3, 3, 0)

R (0, 3, 0)

, and

T = \frac{O + Q}{2} = (\frac{3}{2}, \frac{3}{2}, 0)

. Since

S

is directly above

T

with

T S = 3

, we have

S (\frac{3}{2}, \frac{3}{2}, 3)

.

(A)

O Q = (3, 3, 0)

O S = (\frac{3}{2}, \frac{3}{2}, 3)

cos θ = \frac{O Q \cdot O S}{∣ O Q ∣∣ O S ∣} = \frac{9}{3 2 \cdot \frac{3 3}{2}} = \frac{1}{3} \neq = cos \frac{π}{3}

. False.

(B) The plane through

O

Q

S

has normal

O Q \times O S = (9, - 9, 0) ∥ (1, - 1, 0)

. Equation:

x - y = 0

. True.

(C) Distance from

P (3, 0, 0)

to plane

x - y = 0

\frac{∣3 - 0∣}{2} = \frac{3}{2}

. True.

(D) Line

R S

passes through

R (0, 3, 0)

with direction

R S = (\frac{3}{2}, - \frac{3}{2}, 3)

. Distance from

O

to this line is

\frac{∣ O R \times R S ∣}{∣ R S ∣} = \frac{15}{2}

. True.

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · One or more correct options