# 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 $y=mx+c$ (slope-intercept 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 $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$ for a circle centered at $\left(h,k\right)$.

#### 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: $\left(x-2{\right)}^{2}+\left(y-3{\right)}^{2}={5}^{2}$for a circle centered at $\left(2,3\right)$ with radius $5$.
• Line Equation: $y=2x+1$ (slope-intercept 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: $\left(x-2{\right)}^{2}+\left(2x+1-3{\right)}^{2}={5}^{2}$.
• Solve for $x$ to find possible x-coordinates.
• Using Quadratic Formula: After obtaining the x-coordinates, substitute them back into the line equation to find corresponding y-coordinates.

• If $x$ is $2$, then $y=2×2+1=5$ (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.