I-have-4-collinear-points-A-a-0-B-b-0-C-c-0-and-D-d-0-where-a-b-c-d-gt-0-Find-a-point-E-x-y-such-that-the-following-expression-is-minimised-2-AE-BE-CE-DE-

Question Number 2672 by Yozzi last updated on 24/Nov/15

I h a v e 4 c o l l i n e a r p o i n t s A (a, 0),

B (b, 0), C (c, 0) a n d D (d, 0) w h e r e

\forall a, b, c, d > 0 . F i n d a p o i n t E (x, y) s u c h

t h a t t h e f o l l o w i n g e x p r e s s i o n i s

m i n i m i s e d :

2 (A E + B E + C E + D E) .

Commented by Rasheed Soomro last updated on 24/Nov/15

\forall \underline{a}, b, c, d > 0

T h e r e i s n o a i n v o l v e d i n t h e p r o b l e m .

P o i n t s a r e A (0, 0), B (b, 0), C (c, 0) a n d E (x, y) .

I s t h e r e A (a, 0) i n s t e a d o f A (0, 0) ?

Commented by Yozzi last updated on 24/Nov/15

S o r r y . E r r o r . I m a d e i t A (a, 0) i n s t e a d .

Answered by prakash jain last updated on 24/Nov/15

assume a < b < c < d

Consider 3 points BCE .

BE + CE ⩾ BC .

BE + CE = BC when E is on the same line as BC

and in between them .

b ⩽ x ⩽ d

Similarly for points ADE . a ⩽ x ⩽ d

I am assuming A^{'} s coordinate is (a, 0) .

If a < b < c < d

Then coordinated for point E are

(x, 0) where b ⩽ x ⩽ c (a s s u m i n g a < b < c < d) .