Let E-ABCD be a pyramid on square base ABCD where A is the origin and B and D are lying on positive x-axis and positive y-axis respectively. If E is $(0, 2, 3)$ and $\vec{D E} . (\hat{i} + \hat{j}) = 0$ , then which of the following statement(s) is (are) correct?

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Perpendicular distance of the point D from the plane ABE is $\frac{9}{\sqrt{13}}$

Volume of the tetrahedron ABDE is 2 cubic units

Foot of the perpendicular of the point D in the plane ABE is $(0, \frac{8}{13}, \frac{12}{13})$

Image of the point D in the plane ABE is $(0, \frac{- 10}{13}, \frac{24}{13})$

answer is A, B, C.

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Detailed Solution

$\vec{D E} . (\hat{i} + \hat{j}) = 0$

Foot of perpendicular ot E $(0, 2, 3)$ on xy plane is D(0,2,0)

Plane ABE is $| \begin{matrix} x & y & z \\ 0 & 2 & 3 \\ 2 & 0 & 0 \end{matrix} | = 0$

Question Image

Image of D(0,2,0) in the plane 3y-2z=0 is $(0, \frac{- 10}{13}, \frac{24}{13})$

Foot of perpendiuclar of D(0,2,0) on the plane 3y-2z=0 is $(0, \frac{8}{13}, \frac{12}{13})$

Volume of tetrahedron ABDE= $V = \frac{1}{3} (A r e a o f t r i a n g l e A B D) h e i g h t = \frac{1}{3} (\frac{1}{2} (2) (2)) 3 = 2$

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