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Question Number 136781 by bramlexs22 last updated on 25/Mar/21

What is the equation of the circle passing though (-2,-4) and (4,6) its center lies on the line 3x- 2y+20=0\n
	Answered by EDWIN88 last updated on 26/Mar/21
	$Let P(−2,−4) and Q(4,6) , and midpoint PQ is T(((4−2)/2), ((6−4)/2))=T(1,1). let C(a,b) be a center point the circle. We know that vector PQ ⊥ CT ; then { ((PQ = (4−(−2),(6−(−4))=(6,10))),((CT= (1−a,1−b))) :} PQ.CT = 0 ; 6(1−a)+10(1−b)= 0 ⇒16−6a−10b= 0 ; 3a+5b = 8...(i) Because the center point the circle lies on the line 3x−2y+20 = 0 , we get ⇒ 3a−2b = −20...(ii) Now we can solving for a and b , gives 3a+5b = 8 3a −2b = −20 7b = 28 { ((b=4)),((a= −4)) :} , now we want to compute the radius = ∣CP∣ = (√((−2+4)^2 +(−4−4)^2 )) radius = (√(4+64)) = (√(68.)) Finally the equation of the circle is equal to (x+4)^2 +(y−4)^2 = 68 or x^2 +y^2 +8x−8y−34 = 0$
	$Let P (- 2, - 4) and Q (4, 6), and midpoint$ $PQ is T (\frac{4 - 2}{2}, \frac{6 - 4}{2}) = T (1, 1) .$ $let C (a, b) be a center point the circle .$ $We know that vector PQ ⊥ CT; then$ $Extra close brace or missing open brace$ $Extra close brace or missing open brace$ $Extra close brace or missing open brace$ $B e c a u s e t h e c e n t e r p o i n t t h e c i r c l e l i e s$ $o n t h e l i n e 3 x - 2 y + 20 = 0, w e g e t$ $Extra close brace or missing open brace$ $N o w w e c a n s o l v i n g f o r a a n d b, g i v e s$ $3 a + 5 b = 8$ $3 a - 2 b = - 20$ $7 b = 28 {\begin{cases} b = 4 \\ a = - 4 \end{cases}, n o w w e w a n t t o$ $Extra close brace or missing open brace$ $r a d i u s = \sqrt{4 + 64} = \sqrt{68 .}$ $F i n a l l y t h e e q u a t i o n o f t h e c i r c l e i s e q u a l t o$ $Extra close brace or missing open brace$ $x^{2} + y^{2} + 8 x - 8 y - 34 = 0$
	Commented byEDWIN88 last updated on 26/Mar/21

	Answered by mr W last updated on 26/Mar/21

	$s a y c e n t e r o f c i r c l e i s (u, v), r a d i u s r$ $3 u - 2 v + 20 = 0 . . . (i)$ ${(u + 2)}^{2} + {(v + 4)}^{2} = r^{2} . . . (i i)$ ${(u - 4)}^{2} + {(v - 6)}^{2} = r^{2} . . . (i i i)$ $(i i) - (i i i) :$ $3 u + 5 v - 8 = 0 . . . (i v)$ $(i v) - (i) :$ $7 v - 28 = 0 \Rightarrow v = 4$ $u = \frac{8 - 5 v}{3} = \frac{8 - 20}{3} = - 4$ $r^{2} = {(- 4 + 2)}^{2} + {(4 + 4)}^{2} = 68$ $e q n . o f c i r c l e :$ $Extra close brace or missing open brace$
	Commented byotchereabdullai@gmail.com last updated on 27/Mar/21

	$nice!$
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