
Find the equation of the circle passing through the three points (1,2), (3,-4), (5,-6).
Answer
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Hint: We need to find the equation of the circle passing through the three given points. We start to solve the problem by substituting the three given points in the general equation of the circle. Then, we need to simplify the equations to get the equation of a circle.
Complete step by step solution:
Let the three given points be respectively.
The general form of the equation of the circle is where is the centre of the circle.
We need to substitute the given points in the place of in the equation of a circle.
Substituting the point in the equation of a circle, we get,
Simplifying the above equation, we get,
Rearranging the terms, we get,
Substituting the point in the equation of a circle, we get,
Simplifying the above equation, we get,
Rearranging the terms, we get,
Substituting the point in the equation of a circle, we get,
Simplifying the above equation, we get,
Rearranging the terms, we get,
Multiplying the equation by 2 and adding with the equation , we get,
Multiplying the equation by 3 and adding with the equation , we get,
Multiplying the equation with 3,
Multiplying the equation with 4,
Subtracting the above two equations, we get,
Substituting the value of in the equation , we get,
Shifting to the other side of the equation, we get,
Substituting the value of in the equation , we get,
The equation of the circle is .
The equation of the circle can be diagrammatically represented as follows,
Here, O is the center of the circle.
Note: The result of the given question can be verified by substituting any of the given points in the question. The result attained is correct only if the left-hand side of the equation is equal to the right-hand side of the equation.
Substituting point in the equation, we get,
LHS RHS
The result attained is correct.
Complete step by step solution:
Let the three given points be
The general form of the equation of the circle is
We need to substitute the given points in the place of
Substituting the point
Simplifying the above equation, we get,
Rearranging the terms, we get,
Substituting the point
Simplifying the above equation, we get,
Rearranging the terms, we get,
Substituting the point
Simplifying the above equation, we get,
Rearranging the terms, we get,
Multiplying the equation
Multiplying the equation
Multiplying the equation
Multiplying the equation
Subtracting the above two equations, we get,
Substituting the value of
Shifting
Substituting the value of
The equation of the circle can be diagrammatically represented as follows,
Here, O is the center of the circle.
Note: The result of the given question can be verified by substituting any of the given points in the question. The result attained is correct only if the left-hand side of the equation is equal to the right-hand side of the equation.
Substituting point
The result attained is correct.
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