for-every-real-number-a-b-such-that-a-2-b-2-4a-6b-2-what-is-the-maximum-and-minimum-value-of-the-expression-a-2-b-2-8a-10b-41-

Question Number 79404 by jagoll last updated on 25/Jan/20

for every real number a, b

such that a^{2} + b^{2} - 4 a - 6 b = 2 .

what is the maximum and

minimum value of the

expression

\sqrt{a^{2} + b^{2} - 8 a - 10 b + 41} ?

Commented by john santu last updated on 25/Jan/20

l e t f = \sqrt{a^{2} + b^{2} - 4 a - 6 b - (4 a + 4 b) + 41}

f = \sqrt{43 - (4 a + 4 b)} . T h a t t h e

v a l u e o f f w i l l b e m a x i m u m

i f 4 a + 4 b i s m i n i m u m .

l e t : 4 x + 4 y = l, t h i s i s t h e t a n g e n t

o f c i r c l e {(a - 2)}^{2} + {(b - 3)}^{2} = 15

d = \frac{∣ 8 + 12 - l ∣}{4 \sqrt{2}}, d = r

4 \sqrt{30} = ∣ 20 - l ∣

20 - l = \pm 4 \sqrt{30} \Rightarrow l = 20 \pm 4 \sqrt{30}

(i) l = 20 + 4 \sqrt{30}

f = \sqrt{43 - (20 + 4 \sqrt{30})} = \sqrt{23 - 2 \sqrt{120}}

= \sqrt{15} - 2 \sqrt{2}

(i i) l = 20 - 4 \sqrt{30}

f = \sqrt{43 - (20 - 4 \sqrt{30})} = \sqrt{15} + 2 \sqrt{2}

∴ f_{m a x} = \sqrt{15} + 2 \sqrt{2}

f_{m i n} = \sqrt{15} - 2 \sqrt{2}

Commented by jagoll last updated on 25/Jan/20

great! sir thank you

Commented by peter frank last updated on 25/Jan/20

g r e a t

Commented by john santu last updated on 25/Jan/20

t h a n k s s i r