Base of a pyramid is rectangular, three of its vertices of the base are A (2, 2, -1) B(3, 1, 2) and C (1, 1, 1) and its vertex at the top is P(4,−263,−103) . Fourth vertex of the base is D. Which of the following statements is (are) CORRECT?

P(4,−263,−103) AB¯=i^+j^+3k^,AC¯=−i^−j^+2k^,BC¯=−2i^−k^ AC¯⊥BC¯ A+B2=C+D2⇒D=(4,2,0)Equation of plane through A, B, C, D is x−5y−2z+6=0foot of perpendicular drawn form P is (h, k, r) h−41=k+263−5=r+103−2=−(4+1303+203+6)30 ⇒(h,k,r)=(2,43,23)=(l,m,n)l+m−2n=2, l>m>n volume of pyramid =(13) (Base Area) (height)=(13)(30)(120)=20 cubic units

Base of a pyramid is rectangular, three of its vertices of the base are A (2, 2, -1) B(3, 1, 2) and C (1, 1, 1) and its vertex at the top is P(4,−263,−103) . Fourth vertex of the base is D. Which of the following statements is (are) CORRECT?

P(4,−263,−103) AB¯=i^+j^+3k^,AC¯=−i^−j^+2k^,BC¯=−2i^−k^ AC¯⊥BC¯ A+B2=C+D2⇒D=(4,2,0)Equation of plane through A, B, C, D is x−5y−2z+6=0foot of perpendicular drawn form P is (h, k, r) h−41=k+263−5=r+103−2=−(4+1303+203+6)30 ⇒(h,k,r)=(2,43,23)=(l,m,n)l+m−2n=2, l>m>n volume of pyramid =(13) (Base Area) (height)=(13)(30)(120)=20 cubic units

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Base of a pyramid is rectangular, three of its vertices of the base are A (2, 2, $- 1$ ) B(3, 1, 2) and C (1, 1, 1) and its vertex at the top is $P (4, \frac{- 26}{3}, \frac{- 10}{3})$ . Fourth vertex of the base is D. Which of the following statements is (are) CORRECT?

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If the coordinates of foot of normal drawn from P on the base of the pyramid are $(l, m, n)$ then $l > m > n$ .

Volume of the pyramid is 20 cubic units.

Volume of the pyramid is 60 cubic units.

If the coordinates of foot of normal drawn from P on the base of the pyramid are $(l, m, n)$ then $l + m - 2 n = 2$ .

answer is A, B, C.

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Detailed Solution

$P (4, \frac{- 26}{3}, \frac{- 10}{3})$
$\bar{A B} = \hat{i} + \hat{j} + 3 \hat{k}, \bar{A C} = - \hat{i} - \hat{j} + 2 \hat{k}, \bar{B C} = - 2 \hat{i} - \hat{k}$
$\bar{A C} ⊥ \bar{B C}$
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$\frac{A + B}{2} = \frac{C + D}{2} \Rightarrow D = (4, 2, 0)$
Equation of plane through A, B, C, D is $x - 5 y - 2 z + 6 = 0$
foot of perpendicular drawn form P is (h, k, r)
$\frac{h - 4}{1} = \frac{k + \frac{26}{3}}{- 5} = \frac{r + \frac{10}{3}}{- 2} = \frac{- (4 + \frac{130}{3} + \frac{20}{3} + 6)}{30}$
$\Rightarrow (h, k, r) = (2, \frac{4}{3}, \frac{2}{3}) = (l, m, n)$
$l + m - 2 n = 2, l > m > n$
volume of pyramid $= (\frac{1}{3})$ (Base Area) (height)
$= (\frac{1}{3}) (\sqrt{30}) (\sqrt{120}) = 20$ cubic units