The line y=m x+c and the circle x^2+y^2=a^2 do not intersect...

The line $y=m x+c$ and the circle $x^{2}+y^{2}=a^{2}$ do not intersect if
(a) $a=\frac{|c|}{\sqrt{1+m^{2}}}$
(b) $\mathrm{a}<\frac{|\mathrm{c}|}{\sqrt{1+\mathrm{m}^{2}}}$
(c) $a>\frac{|c|}{\sqrt{1+m^{2}}}$
(d) $a c=\sqrt{1+m^{2}}$

Key Concepts

Distance from a Point to a Line

This concept involves calculating the shortest distance between a fixed point and a line in the coordinate plane using a standard formula. In analytic geometry, the distance from a point (x0, y0) to a line given by Ax + By + C = 0 is determined by the expression |Ax0 + By0 + C| divided by the square root of A² + B². This technique is fundamental for understanding proximity relationships in the plane and is widely applicable in problems concerning tangents and intersections.

Intersection Condition between a Line and a Circle

When analyzing the interaction between a line and a circle, a key approach is to compare the distance from the center of the circle to the line with the radius of the circle. Specifically, if the perpendicular distance from the center to the line is greater than the circle’s radius, the line and circle do not intersect. This criterion is crucial for determining tangency or disjointedness in problems involving circle-line configurations.

Circle Equation and its Properties

The standard equation of a circle in the plane is typically represented in the form x² + y² = a², where 'a' represents the radius and the center is at the origin. This equation is fundamental in analytic geometry as it encapsulates both the geometric definition of a circle and its intrinsic symmetry, which aids in further analysis of intersections and distances with other curves.

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