find-minimum-value-of-function-y-x-6-2-25-x-6-2-121-

Question Number 114922 by bemath last updated on 22/Sep/20

f i n d m i n i m u m v a l u e o f f u n c t i o n

y = \sqrt{{(x + 6)}^{2} + 25} + \sqrt{{(x - 6)}^{2} + 121}

Answered by john santu last updated on 22/Sep/20

y o u w a n t t o f i n d t h e p o i n t o n t h e

x - a x i s s u c h t h a t t h e s u m o f i t s

d i s t a n c e f r o m t h e p o i n t s (- 6, 5) a n d

(6, 11) i s m i n i m a l .

c o n s i d e r t h e s y m e t r i c p o i n t o f (- 6, 5)

w i t h r e s p e c t t o t h e a x i s, t h a t i s (- 6, - 5) .

t h e l i n e t h r o u g h (- 6, - 5) a n d (6, 11)

h a s e q u a t i o n 4 x - 3 y + 9 = 0

a n d i t i n t e r s e c t s t h e x - a x i s a t

x = - \frac{9}{4} . T h e m i n i m u m v a l u e

i s t h e r e f o r e y = \sqrt{{(- \frac{9}{4} + 6)}^{2} + 25} + \sqrt{{(- \frac{9}{6} - 6)}^{2} + 121}

y = \frac{25}{4} + \frac{55}{4} = 20 ∴

Commented by bemath last updated on 22/Sep/20

g a v e k u d o s

Answered by bobhans last updated on 22/Sep/20

w i t h c a l c u l u s

y = \sqrt{{(x + 6)}^{2} + 25} + \sqrt{{(x - 6)}^{2} + 121}

y^{'} = \frac{(x + 6)}{\sqrt{{(x + 6)}^{2} + 25}} + \frac{(x - 6)}{\sqrt{{(x - 6)}^{2} + 121}} = 0

(x + 6) \sqrt{{(x - 6)}^{2} + 121} + (x - 6) \sqrt{{(x + 6)}^{2} + 25} = 0

(x + 6) \sqrt{{(x - 6)}^{2} + 121} = (6 - x) \sqrt{{(x + 6)}^{2} + 25}

\frac{\sqrt{{(x - 6)}^{2} + 121}}{\sqrt{{(x + 6)}^{2} + 25}} = \frac{6 - x}{x + 6}

\frac{{(x - 6)}^{2} + 121}{{(x + 6)}^{2} + 25} = \frac{{(x - 6)}^{2}}{{(x + 6)}^{2}}

{(x^{2} - 36)}^{2} + 121 {(x + 6)}^{2} = {(x^{2} - 36)}^{2} + 25 {(x - 6)}^{2}

{(11 x + 66)}^{2} = {(5 x - 30)}^{2}

(16 x + 36) (6 x + 96) = 0

{\begin{cases} x = - \frac{9}{4} \\ x = - 16 \end{cases}

f o r x = - \frac{9}{4}

y = \sqrt{{(- \frac{9}{4} + 6)}^{2} + 25} + \sqrt{{(- \frac{9}{4} - 6)}^{2} + 121}

y = \sqrt{\frac{625}{16}} + \sqrt{\frac{3025}{16}} = \frac{25}{4} + \frac{55}{4} = 20

f o r x = - 16

y = \sqrt{{(- 16 + 6)}^{2} + 25} + \sqrt{{(- 16 - 6)}^{2} + 121}

y = \sqrt{125} + \sqrt{605} = 35.78

t h e r e f o r e m i n i m u m v a l u e i s 20

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