Four distinct points (0,0),(0,1),(1,0),(2 k, 3 k) lie on a circle. Then the value of k is (a) (1

step 1 So in this problem it is given that four distinct points 0 -0 -1 -10 and 2k comma 3k lie on a ci

Four distinct points (0,0),(0,1),(1,0),(2 k, 3 k) lie on a...

Key Concepts

General Equation of a Circle

In coordinate geometry, the general equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius. This equation is fundamental because it encapsulates the set of all points in the plane that are equidistant from a fixed point. Understanding this form allows one to derive the circle intersecting specific points and forms the basis for many geometric proofs and problem resolutions.

Concyclic Points

Points are said to be concyclic if they all lie on the circumference of the same circle. Determining whether a set of points is concyclic often involves checking if the coordinate conditions can satisfy the circle equation. This concept is significant in geometry because it helps to establish the existence of a circumscribed circle through the points, leading to useful insights in problems involving cyclic quadrilaterals or circle theorems.

System of Equations for Circle Determination

When multiple points are known to lie on a circle, each point can be substituted into the general circle equation to create a system of equations. By solving this system, one can determine unknown parameters of the circle such as the center coordinates and the radius. This method is a powerful tool in coordinate geometry, as it transforms geometric conditions into algebraic equations, which can then be solved using techniques like substitution and elimination.

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