# 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
$${I}\:{have}\:\mathrm{4}\:{collinear}\:{points}\:{A}\left({a},\mathrm{0}\right), \\$$$${B}\left({b},\mathrm{0}\right),\:{C}\left({c},\mathrm{0}\right)\:{and}\:{D}\left({d},\mathrm{0}\right)\:{where}\: \\$$$$\forall{a},{b},{c},{d}>\mathrm{0}.\:{Find}\:{a}\:{point}\:{E}\left({x},{y}\right)\:{such} \\$$$${that}\:{the}\:{following}\:{expression}\:{is} \\$$$${minimised}: \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\left({AE}+{BE}+{CE}+{DE}\right). \\$$
Commented by Rasheed Soomro last updated on 24/Nov/15
$$\forall\underset{−} {{a}},{b},{c},{d}>\mathrm{0} \\$$$$\mathcal{T}{here}\:{is}\:{no}\:\:{a}\:\:{involved}\:{in}\:{the}\:{problem}. \\$$$${Points}\:{are}\:{A}\left(\mathrm{0},\mathrm{0}\right),{B}\left({b},\mathrm{0}\right),\:{C}\left({c},\mathrm{0}\right)\:{and}\:{E}\left({x},{y}\right). \\$$$${Is}\:{there}\:{A}\left({a},\mathrm{0}\right)\:{instead}\:{of}\:{A}\left(\mathrm{0},\mathrm{0}\right)? \\$$
Commented by Yozzi last updated on 24/Nov/15
$${Sorry}.\:{Error}.\:{I}\:{made}\:{it}\:{A}\left({a},\mathrm{0}\right)\:{instead}. \\$$
Answered by prakash jain last updated on 24/Nov/15
$$\mathrm{assume}\:{a}<{b}<{c}<{d} \\$$$$\mathrm{Consider}\:\mathrm{3}\:\mathrm{points}\:\mathrm{BCE}. \\$$$$\mathrm{BE}+\mathrm{CE}\geqslant\mathrm{BC}. \\$$$$\mathrm{BE}+\mathrm{CE}=\mathrm{BC}\:\mathrm{when}\:\mathrm{E}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{same}\:\mathrm{line}\:\mathrm{as}\:\mathrm{BC} \\$$$$\mathrm{and}\:\mathrm{in}\:\mathrm{between}\:\mathrm{them}. \\$$$${b}\leqslant{x}\leqslant{d} \\$$$$\mathrm{Similarly}\:\mathrm{for}\:\mathrm{points}\:\mathrm{ADE}.\:{a}\leqslant{x}\leqslant{d} \\$$$$\mathrm{I}\:\mathrm{am}\:\mathrm{assuming}\:\mathrm{A}'{s}\:\mathrm{coordinate}\:\mathrm{is}\:\left({a},\mathrm{0}\right). \\$$$$\mathrm{If}\:{a}<{b}<{c}<{d} \\$$$$\mathrm{Then}\:\mathrm{coordinated}\:\mathrm{for}\:\mathrm{point}\:\mathrm{E}\:\mathrm{are} \\$$$$\left({x},\mathrm{0}\right)\:\mathrm{where}\:{b}\leqslant{x}\leqslant{c}\:\left({assuming}\:{a}<{b}<{c}<{d}\right). \\$$