Intersection (Geometric) Math Example 4

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Example 4

hard
Find the intersection of the two circles: C1:x2+y2=25C_1: x^2 + y^2 = 25 and C2:(x−4)2+y2=9C_2: (x-4)^2 + y^2 = 9.

Solution

  1. 1
    Step 1: Expand C2C_2: x2−8x+16+y2=9⇒x2+y2=8x−7x^2 - 8x + 16 + y^2 = 9 \Rightarrow x^2 + y^2 = 8x - 7.
  2. 2
    Step 2: Substitute x2+y2=25x^2 + y^2 = 25 from C1C_1: 25=8x−7⇒8x=32⇒x=425 = 8x - 7 \Rightarrow 8x = 32 \Rightarrow x = 4.
  3. 3
    Step 3: Substitute x=4x = 4 into C1C_1: 16+y2=25⇒y2=9⇒y=±316 + y^2 = 25 \Rightarrow y^2 = 9 \Rightarrow y = \pm 3.

Answer

Intersection points: (4,3)(4, 3) and (4,−3)(4, -3).
Subtracting one circle equation from the other eliminates the quadratic terms, yielding the radical axis — the line through both intersection points. Substituting back finds the exact points.

About Intersection (Geometric)

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

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More Intersection (Geometric) Examples

Example 1 easy

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

Example 2 medium

Find the intersection point(s) of line [formula] and circle [formula]. Classify the intersection (se

Example 3 easy

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

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Related Concepts

LineSystems of Equations