Finding Intersection Points of Lines and Circles
Finding intersection points of lines and circles is a Grade 11 algebra skill in Big Ideas Math involving solving a system of a linear and a circular equation. The circle equation (x−h)² + (y−k)² = r² and the line equation y = mx + b are solved simultaneously by substituting the linear expression for y into the circle equation, yielding a quadratic. The discriminant of the quadratic determines the number of intersections: two distinct points (discriminant > 0), one tangent point (discriminant = 0), or no intersection (discriminant < 0). Solutions are the x-values from the quadratic; corresponding y-values come from the line equation.
Key Concepts
To find where a line intersects a circle:.
Step 1. Write the circle equation in standard form $(x h)^2 + (y k)^2 = r^2$ or general form $x^2 + y^2 + Dx + Ey + F = 0$. Step 2. Write the line equation, typically in the form $y = mx + b$ or $x = c$. Step 3. Substitute the linear expression into the circle equation. Step 4. Solve the resulting quadratic equation. Step 5. Find the corresponding coordinates for each solution. Step 6. Write each intersection point as an ordered pair and verify.
Common Questions
How do you find the intersection of a line and a circle?
Substitute the line equation (solve for y) into the circle equation to get a quadratic in x. Solve the quadratic; each real solution gives an x-coordinate. Find y by substituting back into the line.
What does the discriminant tell you about line-circle intersections?
Discriminant > 0: two intersection points (secant). Discriminant = 0: one point of tangency (tangent line). Discriminant < 0: no real intersection (line misses the circle).
How do you solve the system y = 2x + 1 and x² + y² = 25?
Substitute: x² + (2x+1)² = 25 → x² + 4x² + 4x + 1 = 25 → 5x² + 4x − 24 = 0. Solve this quadratic for x, then find y using y = 2x + 1.
What form must the circle equation be in for substitution?
The standard form (x−h)² + (y−k)² = r² or general form x² + y² + Dx + Ey + F = 0 can both be used. Substituting the linear y expression into either form works.
How do you verify that a found point lies on both the line and the circle?
Substitute the (x, y) coordinates into both equations and confirm both are satisfied. This checks algebraic accuracy.
What is a tangent line to a circle?
A tangent line touches the circle at exactly one point (discriminant = 0). It is perpendicular to the radius drawn to the point of tangency.