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.
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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
Thus
and
Let
Note:
If the angle of the intersection of two circles is
The condition for orthogonality is
Example 1: The centre of circle S lies on
1)
2)
3)
4) none of these
Solution
Let
It is of the form
Example 2: If a circle passes through the point
1)
2)
3)
4)
Solution
Let the equation of the circle through
then
Since (i) cuts the circle
so that from (ii), we get
Hence, the answer is the option (1).
Example 3: The circles
1)
2)
3)
4)
Solution
So,
Hence, the answer is the option (4).
Example 4: If two circles
1)
2)
3)
4)
Solution
Centers and radii of the given circles are
from the last two relations,
from first two relations
from eqs. (i) and (ii), we get
Hence, the answer is the option (1).
Example 5: The locus of the centres of the circles which cut the circles
1)
2)
3)
4)
Solution
[Hint: Locus of the centre of the
| Let out circle be
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.