if a circle having an equation x^2 +y^2 −6x−8y=0 is intersected at A and B by x+y=1.find the equation of the circle on AB as diameter


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Question Number 84157 by redmiiuser last updated on 10/Mar/20

$i f a c i r c l e h a v i n g a n$ $e q u a t i o n x^{2} + y^{2} - 6 x - 8 y = 0$ $i s i n t e r s e c t e d a t A a n d B$ $b y x + y = 1 . f i n d t h e$ $e q u a t i o n o f t h e c i r c l e$ $o n A B a s d i a m e t e r$
	Answered by TANMAY PANACEA last updated on 10/Mar/20

	$x^{2} + {(1 - x)}^{2} - 6 x - 8 (1 - x) = 0$ $x^{2} + 1 - 2 x + x^{2} - 6 x - 8 + 8 x = 0$ $2 x^{2} - 7 = 0 x = \pm \sqrt{\frac{7}{2}}$ $A (\sqrt{\frac{7}{2}}, 1 - \sqrt{\frac{7}{2}}) B (- \sqrt{\frac{7}{2}}, 1 + \sqrt{\frac{7}{2}})$ $c e n t r e o f r e q u i r e d c i r c l e m i d p o i n t o f A B$ $(0, 1) d i a m e t e r = d i s t a n c e b e t w e e n A B$ $2 r = {[{(2 \sqrt{\frac{7}{2}})}^{2} + {(2 \sqrt{\frac{7}{2}})}^{2}]}^{\frac{1}{2}}$ $2 r = {[(4 \times \frac{7}{2} + 4 \times \frac{7}{2})]}^{\frac{1}{2}} = 2 \sqrt{7}$ $r = \sqrt{7}$ $e q n c i r c l e$ ${(x - 0)}^{2} + {(y - 1)}^{2} = {(\sqrt{7})}^{2}$ $x^{2} + y^{2} - 2 y - 6 = 0$
	Answered by john santu last updated on 10/Mar/20

	$the equation of circle$ $(x - \sqrt{\frac{7}{2}}) (x + \sqrt{\frac{7}{2}}) + (y - (1 + \sqrt{\frac{7}{2}})) (y - (1 - \sqrt{\frac{7}{2}})) = 0$ $x^{2} - \frac{7}{2} + y^{2} - 2 y - \frac{5}{2} = 0$ $x^{2} + y^{2} - 2 y - 6 = 0$
	Answered by TANMAY PANACEA last updated on 10/Mar/20

	$a n o t h e r w a y$ $t h e e q n o f c i r c l e$ $(x^{2} + y^{2} - 6 x - 8 y) + λ (x + y - 1) = 0$ $x^{2} + y^{2} + x (- 6 + λ) + y (- 8 + λ) - λ = 0$ $c e n t r e o f r e q u i r e d c i r l e i s$ $c o m p a r i n g w i t h x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ $2 g x = (- 6 + λ) x \to 2 g = - 6 + λ g = (\frac{λ - 6}{2})$ $2 f y = (- 8 + λ) y \to 2 f = - 8 + λ f = (\frac{λ - 8}{2})$ $c e n t r e o f r e q u i r e d c i r c l e (- g, - f)$ $(\frac{6 - λ}{2}, \frac{8 - λ}{2}) w h i c h l i e s o n x + y = 1$ $s o \frac{6 - λ}{2} + \frac{8 - λ}{2} = 1 \to 7 - λ = 1 s o λ = 6$ $r e q i r e d c i r c l e$ $x^{2} + y^{2} + x (- 6 + 6) + y (- 8 + 6) - 6 = 0$ $x^{2} + y^{2} - 2 y - 6 = 0$
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