☺☺☺☺please users try to solve this i beg u plz plz plzFind the coordinates of tge centre of circle passing through the points (6,-6),(3,-7),(3,3). Also find the length of its diameter.☺☺☺
Answers
Given circle is passing through three points A(6,-6), B(3,-7) and C(3,3).
Let the centre of the circle is O(x,y). Then OA = OB = OC.
We know that distance formula is : d = √(x2 - x1)^2 + (y2 - y1)^2
(i)
OA = OB
⇒ √(x - 6)^2 + (y+ 6)^2 = √(x - 3)^2 + (y + 7)^2
⇒ x^2 + 36 - 12x + y^2 + 36 + 12y = x^2 + 9 - 6x + y^2 + 49 + 14y
⇒ x^2 + 36 - 12x + y^2 + 36 + 12y - x^2 - 9 + 6x - y^2 - 49 - 14y = 0
⇒ -12x + 12y + 6x - 14y + 72 - 9 - 49 = 0
⇒ 14 - 6x - 2y = 0
⇒ 6x + 2y = 14
⇒ 3x + y = 7 ----- (1)
(ii)
OB = OC
⇒ √(x - 3)^2 + (y + 7)^2 = √(x - 3)^2 + (y - 3)^2
⇒ x^2 + 9 - 6x + y^2 + 49 + 14y = x^2 + 9 - 6x + y^2 + 9 - 6y
⇒ x^2 + 9 - 6x + y^2 + 49 + 14y - x^2 - 9 + 6x - y^2 - 9 + 6y = 0
⇒ 20y + 9 + 49 - 9 - 9 = 0
⇒ 20y + 40 = 0
⇒ 20y = -40
⇒ y = -2 ------- (2)
Now,Substitute (2) in (1), we get
⇒ 3x + y = 7
⇒ 3x + (-2) = 7
⇒ 3x - 2 = 7
⇒ 3x = 9
⇒ x = 3.
Therefore, the centre of the circle going the given points is (3,-2).
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In order to find the the diameter, we have to find the radius 1st.
We know radius is the distance from the centre to any point on circumference.
Radius of the circle OA = √(x - 6)^2 + (y + 6)^2
⇒ √(3 - 6)^2 + (-2 + 6)^2
⇒ √(-3)^2 + (4)^2
⇒ √9 + 16
⇒ √25
⇒ 5.
We know that diameter = 2r.
⇒ 2*5
⇒ 10.
Therefore, the length of its diameter = 10 units.
Hope it helps!
*Centre -> ( 3 , -2 )
*Diameter = 10
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@Yuichi
