a-b-c-abc-a-2-b-2-c-2-49-ab-bc-ca-

Question Number 217735 by ArshadS last updated on 19/Mar/25

a + b + c = abc

a^{2} + b^{2} + c^{2} = 49

ab + bc + ca = ? . .

Commented by mr W last updated on 19/Mar/25

t h e r e i s n o u n i q u e s o l u t i o n .

m a y b e y o u s h o u l d a s k

{(a b + b c + c a)}_{m i n} = ?

o r

{(a b + b c + c a)}_{m a x} = ?

Answered by mr W last updated on 20/Mar/25

a s s u m e a, b, c \in R

s a y u = a + b, v = a b

t h e n w e h a v e

a + b - a b c = - c

\Rightarrow u - c v = - c \dots (i)

{(a + b)}^{2} - 2 a b = 49 - c^{2}

\Rightarrow u^{2} - 2 v = 49 - c^{2} \dots (i i)

w e c a n s o l v e f o r u, v i n t e r m s o f c .

{c u}^{2} - 2 u - c (51 - c^{2}) = 0

\Rightarrow u = \frac{1 \pm \sqrt{1 + c^{2} (51 - c^{2})}}{c}

\Rightarrow v = \frac{u}{c} + 1 = 2 \pm \sqrt{1 + c^{2} (51 - c^{2})}

1 + c^{2} (51 - c^{2}) ⩾ 0

c^{4} - 51 c^{2} - 1 ⩽ 0

\Rightarrow \frac{51 - \sqrt{51^{2} + 4}}{2} < 0 ⩽ c^{2} ⩽ \frac{51 + \sqrt{51^{2} + 4}}{2}

\Rightarrow - \sqrt{\frac{51 + \sqrt{2605}}{2}} ⩽ c ⩽ \sqrt{\frac{51 + \sqrt{2605}}{2}} \approx 7.143

s a y s = a b + b c + c a, t h e n

s = a b + (a + b) c

= v + c u

= 3 \pm 2 \sqrt{1 + c^{2} (51 - c^{2})}

w e s e e s i s n o t u n i q u e .

b u t c^{2} (51 - c^{2}) ⩽ {(\frac{c^{2} + 51 - c^{2}}{2})}^{2} = \frac{51^{2}}{4}

w e h a v e

s = 3 + 2 \sqrt{1 + c^{2} (51 - c^{2})} ⩽ 3 + 2 \sqrt{1 + \frac{51^{2}}{4}}

i . e . s_{m a x} = 3 + 2 \sqrt{1 + \frac{51^{2}}{4}} = 3 + \sqrt{2605}

s i m i l a r l y

s = 3 - 2 \sqrt{1 + c^{2} (51 - c^{2})} ⩾ 3 - 2 \sqrt{1 + \frac{51^{2}}{4}}

i . e . s_{m i n} = 3 - 2 \sqrt{1 + \frac{51^{2}}{4}} = 3 - \sqrt{2605}

s u m m a r y :

w i t h g i v e n c o n d i t i o n s w e c a n n o t

u n i q u e l y d e t e r m i n e a, b, c a n d

a b + b c + c a, b u t w e c a n s a y t h a t

- \sqrt{\frac{51 + \sqrt{2605}}{2}} ⩽ a, b, c ⩽ \sqrt{\frac{51 + \sqrt{2605}}{2}} \approx 7.143

3 - \sqrt{2605} ⩽ a b + b c + c a ⩽ 3 + \sqrt{2605}

Commented by ArshadS last updated on 22/Mar/25

N i c e s i r, t h a n k s a l o t!

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