Intersection (Geometric) Math Example 2

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Example 2

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Find the intersection point(s) of line y=x+3y = x + 3 and circle x2+y2=25x^2 + y^2 = 25. Classify the intersection (secant, tangent, or no intersection).

Solution

  1. 1
    Step 1: Substitute y=x+3y = x+3 into circle: x2+(x+3)2=25โ‡’2x2+6x+9=25โ‡’2x2+6xโˆ’16=0โ‡’x2+3xโˆ’8=0x^2 + (x+3)^2 = 25 \Rightarrow 2x^2 + 6x + 9 = 25 \Rightarrow 2x^2 + 6x - 16 = 0 \Rightarrow x^2 + 3x - 8 = 0.
  2. 2
    Step 2: Discriminant: ฮ”=9+32=41>0\Delta = 9 + 32 = 41 > 0. Two real roots, so two intersection points.
  3. 3
    Step 3: x=โˆ’3ยฑ412x = \dfrac{-3 \pm \sqrt{41}}{2}. Corresponding y=x+3y = x + 3. The line is a secant (crosses the circle twice).

Answer

Two intersections at x=โˆ’3ยฑ412x = \dfrac{-3 \pm \sqrt{41}}{2}, y=x+3y = x+3. The line is a secant.
A line and circle can intersect in 00, 11, or 22 points, determined by the discriminant of the resulting quadratic: negative โ‡’\Rightarrow no intersection, zero โ‡’\Rightarrow tangent, positive โ‡’\Rightarrow secant.

About Intersection (Geometric)

The set of all points where two or more geometric objects (lines, planes, curves) meet or cross each other.

Learn more about Intersection (Geometric) โ†’

More Intersection (Geometric) Examples

Example 1 easy

Find the intersection point of lines [formula] and [formula].

Example 3 easy

Do lines [formula] and [formula] intersect? Explain why or why not.

Example 4 hard

Find the intersection of the two circles: [formula] and [formula].

โ† All Intersection (Geometric) Examples Intersection (Geometric) Concept

Related Concepts

LineSystems of Equations