Equation of a Circle Math Example 4

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Example 4

hard
Determine whether the circles (x−1)2+(y−3)2=16(x-1)^2 + (y-3)^2 = 16 and (x−7)2+(y−3)2=4(x-7)^2 + (y-3)^2 = 4 intersect, and find the number of intersection points.

Solution

  1. 1
    Circle 1: center (1,3)(1,3), r1=4r_1=4. Circle 2: center (7,3)(7,3), r2=2r_2=2. Distance between centers: d=(7−1)2+0=6d = \sqrt{(7-1)^2 + 0} = 6.
  2. 2
    Check: ∣r1−r2∣=2<d=6<r1+r2=6|r_1 - r_2| = 2 < d = 6 < r_1 + r_2 = 6. Since d=r1+r2d = r_1 + r_2, the circles are externally tangent — they have exactly 1 intersection point.

Answer

Exactly 1 intersection point (externally tangent)\text{Exactly 1 intersection point (externally tangent)}
Two circles can have 0, 1, or 2 intersection points. Compare the distance dd between centers to r1+r2r_1 + r_2 and ∣r1−r2∣|r_1 - r_2|: if d>r1+r2d > r_1+r_2 or d<∣r1−r2∣d < |r_1-r_2|, no intersection; if d=r1+r2d = r_1+r_2 or d=∣r1−r2∣d = |r_1-r_2|, tangent (1 point); if ∣r1−r2∣<d<r1+r2|r_1-r_2| < d < r_1+r_2, two intersection points.

About Equation of a Circle

The standard form equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h,k)(h, k) and radius rr in the coordinate plane.

Learn more about Equation of a Circle →

More Equation of a Circle Examples

Example 1 easy

Write the equation of the circle with center [formula] and radius [formula].

Example 2 medium

Find the center and radius of the circle [formula].

Example 3 medium

Find the equation of the circle that has a diameter with endpoints [formula] and [formula].

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