Intersection of a Line and a Circle
Intersection of a Line and a Circle:
1. Equation of a Line:
 A line is represented in the Cartesian plane by the equation $y=mx+c$ (slopeintercept form) or $Ax+By+C=0$ (general form).
2. Equation of a Circle:
 The equation of a circle is ${x}^{2}+{y}^{2}={r}^{2}$ for a circle centered at the origin or $(xh{)}^{2}+(yk{)}^{2}={r}^{2}$ for a circle centered at $(h,k)$.
3. Intersection Conditions:
 Geometric Concept: The intersection of a line and a circle occurs when the line passes through, is tangent to, or doesn’t touch the circle at all.
 Mathematical Condition: For intersection, the coordinates satisfying both the line equation and the circle equation simultaneously represent the intersection points.
4. Solving for Intersection Points:

Example Equations:
 Circle Equation: $(x2{)}^{2}+(y3{)}^{2}={5}^{2}$for a circle centered at $(2,3)$ with radius $5$.
 Line Equation: $y=2x+1$ (slopeintercept form).

Substitution Method: Substitute $y=2x+1$ into the circle equation to form a quadratic equation in terms of $x$.
 Substitute $y$ in the circle equation: $(x2{)}^{2}+(2x+13{)}^{2}={5}^{2}$.
 Solve for $x$ to find possible xcoordinates.

Using Quadratic Formula: After obtaining the xcoordinates, substitute them back into the line equation to find corresponding ycoordinates.
 If $x$ is $2$, then $y=2\times 2+1=5$ (one intersection point).
5. Geometric Interpretation:
 Visual Analysis: Plotting the line and circle on a graph helps visualize their intersection points or lack thereof.
 Understanding Tangency: A line tangent to a circle intersects it at a single point, whereas a line passing through the circle has two intersection points.
6. Special Cases:
 No Intersection: If the discriminant of the quadratic equation (${b}^{2}4ac$) is negative, there is no real intersection (line doesn’t intersect the circle).
 Tangency: A line tangent to the circle has a single point of intersection with the circle.