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Question Number 103945 by bobhans last updated on 18/Jul/20

a c u b e A B C D . E F G H w i t h l e n g t h s i d e

4 c m . G i v e n p o i n t P i s m i d p o i n t E F .

f i n d t h e d i s t a n c e o f l i n e A P t o l i n e

H B .

Answered by bramlex last updated on 18/Jul/20

l e t : a x + b y + c z = - 1 b e a

e q u a t i o n o f p l a n e H B P^{'} w h e r e

B P^{'} p a r a l l e l t o A P .

c o o r d i n a t e s A (4, 0, 0); B (4, 4, 0)

; P^{'} (4, 6, 4) a n d H (0, 0, 4)

(i) s u b s t i t u t e B \Rightarrow 4 a + 4 b = - 1

(i i) s u b s t i t u t e H \Rightarrow 4 c = - 1 \Rightarrow c = - \frac{1}{4}

(i i i) s u b s t i t u t e P

^{'} \Rightarrow 4 a + 6 b + 4 c = - 1 \Rightarrow 4 a + 6 b = 0

s u b s t r a c t (i) & (i i i)

w e g e t 2 b = 1 \Rightarrow b = \frac{1}{2} & a = - \frac{3}{4}

s o e q o f p l a n e H B P^{'} \Rightarrow - \frac{3}{4} x + \frac{1}{2} y - \frac{1}{4} z + 1 = 0

o r - 3 x + 2 y - z + 4 = 0

s o t h e d i s t a n c e w e d e s i r e d i s \frac{∣ - 12 + 4 ∣}{\sqrt{9 + 4 + 1}}

= \frac{8}{\sqrt{14}} = \frac{4 \sqrt{14}}{7} c m . ◼

Commented by bramlex last updated on 18/Jul/20

Commented by bobhans last updated on 18/Jul/20

w a w \dots n i c e a n d c o o l l

Answered by mr W last updated on 18/Jul/20

Commented by mr W last updated on 18/Jul/20

A (4, 0, 4)

P (4, 2, 0)

H (0, 0, 0)

B (4, 4, 4)

e q n . o f A P = (4, 0, 4) + λ (0, 1, - 2)

e q n . o f H B = (0, 0, 0) + μ (1, 1, 1)

a p o i n t o n A P i s S (4, λ, 4 - 2 λ)

a p o i n t o n H B i s T (μ, μ, μ)

l e t Φ = {S T}^{2}

Φ = {(4 - μ)}^{2} + {(λ - μ)}^{2} + {(4 - 2 λ - μ)}^{2}

\frac{\partial Φ}{\partial s} = 2 (λ - μ) + 4 (2 λ + μ - 4) = 0

\Rightarrow 5 λ + μ = 8 \dots (i)

\frac{\partial Φ}{\partial t} = - 2 (4 - μ) - 2 (λ - μ) + 2 (2 λ + μ - 4) = 0

\Rightarrow λ + 3 μ = 8 \dots (i i)

\Rightarrow λ = \frac{8}{7}

\Rightarrow μ = \frac{16}{7}

Φ_{m i n} = {(4 - \frac{16}{7})}^{2} + {(\frac{8}{7} - \frac{16}{7})}^{2} + {(4 - \frac{16}{7} - \frac{16}{7})}^{2} = \frac{224}{49}

m i n . d i s t a n c e f r o m A P t o H B i s

{S T}_{m i n} = \sqrt{Φ_{m i n}} = \sqrt{\frac{224}{49}} = \frac{4 \sqrt{14}}{7}

Commented by bobhans last updated on 18/Jul/20

w h a t m e t h o d i t i s s i r ?

Commented by bobhans last updated on 18/Jul/20

w h e r e b e c o m e s (0, 1, - 2) s i r

Commented by mr W last updated on 18/Jul/20

\vec{A P} = (4 - 4, 2 - 0, 0 - 4) = (0, 2, - 4) o r

(0, 1, - 2)

Commented by bramlex last updated on 18/Jul/20

w a w \dots g r e a t y o u r m e t h o d s i r .

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