Find-the-equation-of-the-circle-circumscribing-the-triangle-form-by-lines-3x-y-5-0-x-y-1-0-and-2x-y-4-0-

Question Number 6192 by sanusihammed last updated on 17/Jun/16

F i n d t h e e q u a t i o n o f t h e c i r c l e c i r c u m s c r i b i n g t h e t r i a n g l e f o r m b y

l i n e s 3 x + y - 5 = 0, x + y + 1 = 0, a n d 2 x + y - 4 = 0

Answered by Rasheed Soomro last updated on 18/Jun/16

3 x + y - 5 = 0 \dots \dots (i)

x + y + 1 = 0 \dots \dots . . (i i)

2 x + y - 4 = 0 \dots \dots . (i i i)

(i) - (i i) : 2 x - 6 = 0 \Rightarrow x = 3 \Rightarrow y = - 4 : (3, - 4)

(i i i) - (i i) : x - 5 = 0 \Rightarrow x = 5 \Rightarrow y = - 6 : (5, - 6)

(i) - (i i i) : x - 1 = 0 \Rightarrow x = 1 \Rightarrow y = 2 : (1, 2)

L e t e q u a t i o n p a s s i n g t h r o u g h v e r t i c e s

(3, - 4), (5, - 6) a n d (1, 2) i s :

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

S i n c e (3, - 4), (5, - 6) a n d (1, 2) l i e o n t h e

c i r c l e, {t h e y}^{'} l l s a t i s f y t h e e q u a g i o n

{(3)}^{2} + {(- 4)}^{2} + 2 g (3) + 2 f (- 4) + c = 0

6 g - 8 f + c = - 25 \dots \dots \dots \dots \dots . I

{(5)}^{2} + {(- 6)}^{2} + 2 g (5) + 2 f (- 6) + c = 0

10 g - 12 f + c = - 61 \dots \dots \dots . . I I

{(1)}^{2} + {(2)}^{2} + 2 g (1) + 2 f (2) + c = 0

2 g + 4 f + c = - 5 \dots \dots \dots \dots \dots I I I

I I - I : 4 g - 4 f = - 36 \Rightarrow g - f = - 9 \dots \dots \dots . . A

I I - I I I : 8 g - 16 f = - 56 \Rightarrow g - 2 f = - 7 \dots \dots B

A - B : f = - 2, A : g - (- 2) = - 9 \Rightarrow g = - 11

I I I : 2 (- 11) + 4 (- 2) + c = - 5 \Rightarrow c = 25

T h e r e q u i r e d e q u a t i o n :

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

\Rightarrow x^{2} + y^{2} + 2 (- 11) x + 2 (- 2) y + (25) = 0

\Rightarrow x^{2} + y^{2} - 22 x - 4 y + 25 = 0

Commented by sanusihammed last updated on 18/Jun/16

T h a n k s s o m u c h

Commented by sanusihammed last updated on 18/Jun/16

i r e a l l y a p p r e c i a t e