If the lines a x+b y+c1=0 and a x+b y+c2=0,(c1 c2 ≠0) intersect the co-ordinate axes at con-cycl

step 1 So in this question it's given that these two straight lines intersect the coordinate axis at co

If the lines a x+b y+c1=0 and a x+b y+c2=0,(c1 c2 ≠0)...

If the lines $a x+b y+c_{1}=0$ and $a x+b y+c_{2}=0,\left(c_{1} c_{2} \neq 0\right)$ intersect the co-ordinate axes at con-cyclic points, then
(a) $a=b$
(b) $a^{2}=b^{2}$
(c) $\mathrm{ac}_{1}=\mathrm{bc}_{2}$
(d) $\mathrm{ac}_{1}+\mathrm{bc}_{2}=0$

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Key Concepts

Intercepts of a Linear Equation

In coordinate geometry, given a linear equation in the form ax + by + c = 0, the points where the line crosses the coordinate axes are called the intercepts. The x-intercept is found by setting y = 0 (resulting in x = -c/a, assuming a ? 0) and the y-intercept is found by setting x = 0 (yielding y = -c/b, assuming b ? 0). These intercepts are crucial for analyzing geometric properties involving the line, such as determining where multiple lines intersect the axes.

Concyclic Points

A set of points are said to be concyclic if there exists a single circle that passes through all of them. In the context of coordinate geometry, when points (such as the intercepts of lines) are required to be concyclic, they must satisfy specific algebraic conditions derived from the general equation of a circle. This concept is key to problems involving circles defined by points associated with linear equations.

Equation of a Circle

The general equation of a circle in the plane is given by x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. When multiple points are known to lie on a circle, substituting their coordinates into this equation yields a system of equations. Solving these can provide the necessary conditions for certain geometric configurations, such as the four intercepts from two lines being concyclic.

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