If A-B-C and D be four distinct points in space such that AB is not perpendicular to CD and satisfies AB-CD-1-AD2-BC2-AC2-BD2- then the value of - is

Let O be the origin and a,b,c and d be the position vector of A,B,C and D. AB·CD=(b a)·(d c) ⇒AB·CD=b·d+c·a b·c d·a ⋯(i) ⇒((AD)^2+(BC)^2 (AC)^2 (BD)^2)=(d a) ...

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Standard XII

Mathematics

Condition for a Line to Lie on a Plane

If A,B,C and ...

Question

If $A, B, C$ and $D$ be four distinct points in space such that $- - \to A B$ is not perpendicular to $- - \to C D$ and satisfies $- - \to A B \cdot - - \to C D = \frac{1}{λ} ((- - \to A D)^{2} + (- - \to B C)^{2} - (- - \to A C)^{2} - (- - \to B D)^{2}),$ then the value of $λ$ is

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Solution

Let $O$ be the origin and $\to a, \to b, \to c$ and $\to d$ be the position vector of $A, B, C$ and $D .$
$- - \to A B \cdot - - \to C D = (\to b - \to a) \cdot (\to d - \to c)$
$\Rightarrow - - \to A B \cdot - - \to C D = \to b \cdot \to d + \to c \cdot \to a - \to b \cdot \to c - \to d \cdot \to a \dots (i)$

$\Rightarrow ((- - \to A D)^{2} + (- - \to B C)^{2} - (- - \to A C)^{2} - (- - \to B D)^{2}) = (\to d - \to a)^{2} + (\to c - \to b)^{2} - (\to c - \to a)^{2} - (\to d - \to b)^{2}$
$= | \to d |^{2} + | \to a |^{2} - 2 \to d \cdot \to a + | \to c |^{2} + | \to b |^{2} - 2 \to c \cdot \to b - | \to c |^{2} - | \to a |^{2} + 2 \to c \cdot \to a - | \to d |^{2} - | \to b |^{2} + 2 \to d \cdot \to b$
$= 2 (\to b \cdot \to d + \to c \cdot \to a - \to b \cdot \to c - \to d \cdot \to a)$
Using $(i),$ we have
$λ = 2$

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If A,B,C and D be four distinct points in space such that −−→AB is not perpendicular to −−→CD and satisfies −−→AB⋅−−→CD=1λ((−−→AD)2+(−−→BC)2−(−−→AC)2−(−−→BD)2), then the value of λ is

If $A, B, C$ and $D$ be four distinct points in space such that $- - \to A B$ is not perpendicular to $- - \to C D$ and satisfies $- - \to A B \cdot - - \to C D = \frac{1}{λ} ((- - \to A D)^{2} + (- - \to B C)^{2} - (- - \to A C)^{2} - (- - \to B D)^{2}),$ then the value of $λ$ is