The centres of the three circles x^2+y^2-4 x-6 y-12=0,...

The centres of the three circles $x^{2}+y^{2}-4 x-6 y-12=0, x^{2}+y^{2}+2 x+4 y-10=0, x^{2}+y^{2}-10 x-16 y-1=0$ are
(a) collinear
(b) non-collinear
(c) collinear and lie on $\mathrm{x}$ -axis
(d) collinear and lie on $\mathrm{y}=\mathrm{x}+1$

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Key Concepts

Collinearity

Collinearity is a concept in geometry where three or more points lie on the same straight line. Verifying collinearity often involves showing that the slopes between consecutive pairs of points are equal.

Slope Calculation

Calculating the slope involves determining the ratio of the change in the y-coordinates to the change in the x-coordinates between two points. Consistent slope values between different pairs of points indicate that the points are aligned along a single straight line.

Standard Equation of a Circle

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center and r is the radius. This form makes it straightforward to identify the circle's key characteristics.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic equations in a form that reveals key features. In the context of circles, it is used to transform the general equation into the standard form, thereby easily determining the center and radius.

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