given-x-y-is-a-point-on-circle-x-2-y-2-6x-4y-23-0-find-minimum-and-maximum-value-of-4x-3y-

Question Number 79395 by jagoll last updated on 24/Jan/20

given (x, y) is a point on circle

x^{2} + y^{2} - 6 x + 4 y - 23 = 0 .

find minimum and maximum

value of 4 x + 3 y

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

\Rightarrow {(x - 3)}^{2} + {(y + 2)}^{2} = 23 + 9 + 4

{(x - 3)}^{2} + {(y + 2)}^{2} = 36

c e n t e r a t (3, - 2), r = 6

\Rightarrow l e t 4 x + 3 y = k, t h i s i s a t a n g e n t

t o c i r c l e .

d = \frac{∣ 4.3 + (- 2) . 3 - k ∣}{\sqrt{4^{2} + 3^{2}}}, d = r

30 = ∣ 6 - k ∣ \Rightarrow 6 - k = \pm 30

n o w w e g e t k_{m i n} = - 30

a n d k_{m a x} = 30

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

t h a n k y o u