If the lines a1 x+b1 y+c1=0 and a2 x+b2 y+c2=0 cut the...

If the lines $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$ cut the coordinate axes in concyclic points, then
(A) $a_{1} a_{2}=b_{1} b_{2}$
(B) $a_{1} b_{1}=a_{2} b_{2}$
(C) $a_{1} b_{2}=a_{2} b_{1}$
(D) none of these

Key Concepts

Concyclic Points

Concyclic points are points that all lie on the same circle. In coordinate geometry, three non?collinear points uniquely determine a circle, so if a fourth point is to be on that circle as well, its coordinates must satisfy the circle's equation. This concept is widely used to derive conditions or constraints when multiple points are required to be concyclic.

Intercepts of a Line

The intercepts of a line are the points where the line crosses the coordinate axes. These are found by setting y = 0 to obtain the x-intercept and x = 0 for the y-intercept. Expressing these intercepts in terms of the line's coefficients is a fundamental part of analytic geometry and is often used to relate the geometry of the line to algebraic conditions.

Circle Equation in Cartesian Coordinates

The general equation of a circle in Cartesian coordinates is given by x² + y² + 2gx + 2fy + c = 0. This form is useful for analyzing geometric properties of the circle such as its center and radius. When points are known to lie on the circle, substituting their coordinates into this equation leads to specific algebraic conditions among the parameters.

Analytic Conditions for Concyclicity

Analytic geometry provides methods to test concyclicity by substituting point coordinates into the general equation of a circle or by setting up a determinant condition. These techniques allow one to derive algebraic relationships among the given coefficients or parameters that must hold for a set of points to be concyclic.

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