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Angle of Intersection of Two Circle: How to Find, Formula, Derivation

Angle of Intersection of Two Circle: How to Find, Formula, Derivation

author
Komal MiglaniUpdated on 02 Jul 2025, 07:40 PM IST

The angle of intersection between two circles is a geometric concept that describes the angle formed by the tangents to the circles at their points of intersection. Understanding the angle of intersection enhances our knowledge of circle geometry and its practical implications.

This Story also Contains

  1. The angle of the Intersection of Two Circle
  2. Recommended Video Based on Angle of Intersection of Two Circle
  3. Solved Examples Based on Angle of Intersection of Two Circle
  4. Summary
Angle of Intersection of Two Circle: How to Find, Formula, Derivation
Angle of Intersection of Two Circle: How to Find, Formula, Derivation

The angle of the Intersection of Two Circle

The angle of Intersection of Two circles is defined as the angle between the tangents drawn to both circles at their point of intersection.

Let the equation of two circle be

S1:x2+y2+2g1x+2f1y+c1=0S2:x2+y2+2g2x+2f2y+c2=0

C1 and C2 are the centres of the given circles and r1 and r2 are the radii of the circles.

Thus C1=(g1,f1) and C2=(g2,f2)

r1=g12+f12c1

and r2=g22+f22c2
Let

d=C1C2=(g1g2)2+(f1f2)2=g12+g22+f12+f222(g1g2+f1f2)
 In ΔC1PC2,cosα=(r12+r22d22r1r2)cos(180θ)=(r12+r22d22r1r2)[α+θ+90+90=360]
cosθ=|r12+r22d22r1r2|

Note:
If the angle of the intersection of two circles is 90, then the circles are said to be orthogonal circles.

The condition for orthogonality is 2(g1g2+f1f2)=c1+c2

Recommended Video Based on Angle of Intersection of Two Circle


Solved Examples Based on Angle of Intersection of Two Circle

Example 1: The centre of circle S lies on 2x2y+9=0 and it cuts orthogonally the circle x2+y2=4. Then the circle passes through two fixed points
1) (1,1),(3,3)
2) (12,12),(4,4)
3) (0,0),(5,5)
4) none of these

Solution
Let S=x2+y2+2gx+2fy+c=0
: it cuts x2+y2=4orthogonally 

2g1g2+2f1f2=c1+c2c=4

(g,f) lies on 2x2y+9=0

2g+2f+9=0Sx2+y2+2gx+2fy+4=0x2+y2+(2f+9)x+2fy+4=0(x2+y2
It is of the form S+λP=0 and hence passes through the II y

Example 2: If a circle passes through the point (a,b) and cuts the circle x2+y2=k2 orthogonally, equation of the locus of its centre is
1) 2ax+2by=a2+b2+k2
2) ax+by=a2+b2+k2
3) x2+y2+2ax+2by+k2=0
4) x2+y22ax2by+a2+b2k2=0

Solution
Let the equation of the circle through (a,(B) be

x2+y2+2gx+2fy+c=0

then a2+b2+2ga+2fb+c=0
Since (i) cuts the circle x2+y2=k2 orthogonally, we have

2 g×0+2f×0=ck2c=k2

so that from (ii), we get a2+b2+2ga+2fb+k2=0, and the locus of the center of (i) is 2ax+2by(a2+b2+k2)=0

Hence, the answer is the option (1).

Example 3: The circles x2+y210x+9=0 and x2+y2=r2 intersect each other in two distinct points if
1) r>8
2) r<2
3) 7<r<11
4) 1<r<9

Solution

C1(5,0),r1=4C2(0,0),r2=r(C1C2)=5
So, r4<5<r+4

r<g&r>11<r<9
Hence, the answer is the option (4).

Example 4: If two circles (x1)2+(y3)2=r2 and x2+y28x+2y+8=0 intersect in two distinct points, then
1) 2<r<8
2) r<2
3) r=2
4) r>2

Solution
Centers and radii of the given circles are C1(1,3),r1=r and C2=(4,1),r2=3 respectively since circles intersect in two distinct points, then

|r1r2|<C1C2<r1+r2|r3|<5<r+3

from the last two relations, r>2
from first two relations

|r3|<55<r3<52<r<8

from eqs. (i) and (ii), we get 2<r<8
Hence, the answer is the option (1).

Example 5: The locus of the centres of the circles which cut the circles x2+y2+4x6y+9=0 and x2+y25x+4y2=0 orthogonally is
1) 9x+10y7=0
2) xy+2=0
3) 9x10y+11=0
4) 9x+10y+7=0

Solution

[Hint: Locus of the centre of the cutting S1=0 and S2=0 orthogonally is the radical axis between S1=0 and S2=0 ]

| Let out circle be x2+y2+2gx+2fy+c=0 conditions 2(g)(2)+2(f)(3)=c+9 and 2(g)(5/2)+2(f)(2)=c2 ag10f=11
locus of centre 9x10y+11=0 Hence, the answer is the option(3).

Summary

The angle of intersection between two circles provides valuable insights into the geometric relationship between them. By using the formulas and understanding the properties of the circles, one can determine the angle formed by the tangents at the points of intersection. This concept is applicable in various fields, including engineering, computer graphics, astronomy, and mathematical problem-solving.