Answer:
<em>The equation of the circle passing through the points: (1, 7) (8, 6) (7, -1) is x^2+y^2-8x-6y=0</em>
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Step-by-step explanation:
<em>In principle there are multiple ways one can find the equation of a circle given 3 points</em>. For this case, we shall use the most common one, i.e. the General Equation of a Circle. In theory the equation actually reads as:
where is the radius, whilst and are points on the circle. Now since here the information given is not as such we shall use the alternate equation of the Circle passing through points, given as:
Eqn (1).
where , and are constant coefficients that can be solved by plugging in the points given (x,y) and creating a System of Linear Equations as follow:
,
Eqn (2)
Eqn (3)
Eqn (4)
Now we have a system of three linear equations (i.e. Eqns (2), (3) and (4)) that can be solved to find the constants D, E and F as follow.
Lets start by subtracting Eqn (3) from (2) so that:
which solving for E we have Eqn. (5)
Lets do the same and subtract Eqn(4) from (2) so that:
which solving for E we have Eqn. (6)
<em>Equating Eqns (5) and (6) we then obtain:</em>
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and thus plugging in D in Eqn. (6) we have:
Finally we can find F using our D and E values and any of the Eqns (2),(3) or (4) as follow (using Eqn (1)):
So finally we have that so our equation of the circle will eventually read: