Intersection of Line and Circle

Intersection of Line and Circle#

Line and Circle Equations Revision#

Theory#

Before studying intersections, let’s review the key equations:

  • Line: \(ax + by + c = 0\) or \(y = mx + k\)

  • Circle: \((x - h)^2 + (y - k)^2 = r^2\) or \(x^2 + y^2 + 2gx + 2fy + c = 0\)

  • Intersection points satisfy both equations simultaneously

Application#

For line \(y = x + 1\) and circle \(x^2 + y^2 = 25\):

  • Substitute the line equation into the circle equation

  • Solve the resulting quadratic equation

Intersection of Line and Circle#

Theory#

A line and circle can intersect in three ways:

  1. Two points: Line is a secant

  2. One point: Line is a tangent

  3. No points: Line misses the circle

Finding Intersection Points#

To find intersections:

  1. Express line in form \(y = mx + c\) (or \(x = k\) for vertical)

  2. Substitute into circle equation

  3. Solve resulting quadratic equation

Discriminant Analysis#

For the quadratic \(Ax^2 + Bx + C = 0\):

  • \(\Delta = B^2 - 4AC > 0\): Two intersections (secant)

  • \(\Delta = B^2 - 4AC = 0\): One intersection (tangent)

  • \(\Delta = B^2 - 4AC < 0\): No intersections

Distance from Center to Line#

For line \(ax + by + c = 0\) and circle center \((h, k)\): $\(d = \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}}\)$

  • If \(d < r\): Line intersects circle (secant)

  • If \(d = r\): Line touches circle (tangent)

  • If \(d > r\): Line misses circle

Interactive Visualization: Line-Circle Intersection Explorer#

Application#

Examples#

Example 1: Finding Intersection Points Find where the line \(y = x + 2\) intersects the circle \(x^2 + y^2 = 10\).

Solution: Substitute \(y = x + 2\) into the circle equation: \(x^2 + (x + 2)^2 = 10\) \(x^2 + x^2 + 4x + 4 = 10\) \(2x^2 + 4x - 6 = 0\) \(x^2 + 2x - 3 = 0\) \((x + 3)(x - 1) = 0\)

So \(x = -3\) or \(x = 1\)

When \(x = -3\): \(y = -3 + 2 = -1\) When \(x = 1\): \(y = 1 + 2 = 3\)

Intersection points: \((-3, -1)\) and \((1, 3)\)

Example 2: Determining Tangency Show that the line \(3x + 4y = 25\) is tangent to the circle \(x^2 + y^2 = 25\).

Solution: The circle has center \((0, 0)\) and radius \(r = 5\).

Distance from center to line: \(d = \frac{|3(0) + 4(0) - 25|}{\sqrt{3^2 + 4^2}} = \frac{25}{\sqrt{25}} = \frac{25}{5} = 5\)

Since \(d = r = 5\), the line is tangent to the circle.

Example 3: Finding Tangent Lines Find equations of tangent lines to circle \((x - 2)^2 + (y - 3)^2 = 5\) with slope \(m = 2\).

Solution: Line: \(y = 2x + c\), or \(2x - y + c = 0\)

For tangency, distance from center \((2, 3)\) to line equals radius \(\sqrt{5}\): \(\frac{|2(2) - 3 + c|}{\sqrt{4 + 1}} = \sqrt{5}\) \(\frac{|1 + c|}{\sqrt{5}} = \sqrt{5}\) \(|1 + c| = 5\)

So \(c = 4\) or \(c = -6\)

Tangent lines: \(y = 2x + 4\) and \(y = 2x - 6\)

Multiple Choice Questions#

Sector Specific Questions: Line-Circle Intersection Applications#

Key Takeaways#

Important

Line-Circle Intersection - Essential Concepts

  1. Types of Intersection:

    • Secant: 2 points (line crosses circle)

    • Tangent: 1 point (line touches circle)

    • No intersection (line misses circle)

  2. Methods to Determine Intersection:

    • Algebraic: Solve simultaneous equations

    • Geometric: Compare distance from center to line with radius

  3. Distance Formula: For line \(ax + by + c = 0\) and point \((x_0, y_0)\): $\(d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}\)$

  4. Discriminant Test: For quadratic from substitution:

    • \(\Delta > 0\): Two intersections

    • \(\Delta = 0\): One intersection (tangent)

    • \(\Delta < 0\): No intersection

  5. Applications: Clearance calculations, optical paths, coverage analysis, lighting design

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