If the curves x^2+y^2-8|x|-9=0 and y=|x|+c do not intersect,...

If the curves $x^{2}+y^{2}-8|x|-9=0$ and $y=|x|+c$ do not intersect, then
(a) $c<-4-5 \sqrt{2}$ or $c>-4+5 \sqrt{2}$
(b) $-4-5 \sqrt{2}<\mathrm{c}<-4+5 \sqrt{2}$
(c) $-5 \sqrt{2}<\mathrm{c}<5 \sqrt{2}$
(d) $-4<c<4$

Key Concepts

Absolute Value Function

The absolute value function transforms an expression into its non-negative value, rendering equations piecewise-defined depending on the sign of the input. In many problems, it creates symmetry about the y-axis, and the function's behavior splits the domain into distinct cases that must be analyzed separately to fully grasp the overall behavior of the graph.

Graph Transformations and Conic Sections

Completing the square and other algebraic manipulations allow for the transformation of quadratic equations into standard forms, which is essential in identifying and interpreting conic sections such as circles. When absolute value terms appear in these equations, the resulting graph may represent multiple conic sections or a piecewise-defined curve, requiring careful analysis for each interval.

Intersection Conditions of Curves

Determining the intersection of curves involves finding solutions that satisfy both equations simultaneously. This often reduces to solving a quadratic equation, where conditions for non-intersection or tangency are derived by setting the discriminant to be negative or zero, respectively. The analysis of these conditions, especially in the presence of parameters, is critical for understanding whether and how the curves interact.

Parameter Analysis in Equations

Introducing a parameter into an equation, such as the constant c modifying the position of a line, leads to a family of curves. Analyzing the impact of this parameter on the relative positions of the curves helps determine the range of values that result in no intersection. This kind of parameter study is key in problems where the aim is to find conditions for non-intersection or tangency between curves.

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