Question

Equation of circle passing through
non-collinear points A, B, C is​

Answer 1

Answer:

Given: Three non-collinear points A, B and C

To prove: One and only one circle can be drawn through A, B, and C

Construction: Join AB and BC. Draw perpendicular bisectors of AB and BC. Let these perpendicular bisectors meet at a point O.

From above it follows that a unique circle passing through 3 points can be drawn given that the points are non-collinea.

Answer 2

Concept:

Equation of a circle:

A unique circle will pass through a given set of three non-collinear points.

The general equation of the circle is

$x^2 + y^2 + 2gx+2fy+c = 0$

Where g, f, and c are constants specific to a circle.

Given:

Points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃).

Find:

The circle which passes through points A, B, and C.

Solution:

Assume the equation of the required circle is

$x^2 + y^2 + 2gx+2fy+c = 0$

As the circle passes through each point A, B, and C, when we substitute the coordinates of these points should satisfy the equation.

⇒ we obtain three equations as follows:

${x_{1}}^2 + {y_{1}}^2 + 2g{x_{1}}+2f{y_{1}}+c = 0$ ---------(1)

${x_{2}}^2 + {y_{2}}^2 + 2g{x_{2}}+2f{y_{2}}+c = 0$ ---------(2)

${x_{3}}^2 + {y_{3}}^2 + 2g{x_{3}}+2f{y_{3}}+c = 0$ ----------(3)

As(x₁, y₁), (x₂, y₂), and (x₃, y₃) are known, the above equations will form a system of linear equations with g, f, and c as variables.

Solving the above system of linear equations will give the unknown values in the equation of the circle g, f, and c.

Finally substituting the values of g, f, and c obtained in the general equation of a circle we will obtain the equation of circle passing through the three points.

∴ The equation of the required circle passing through three non-collinear points can be obtained by solving the obtained linear equations and finding the values of g, f, and c.

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