Prove that two circles cannot intersect at more than two points.
Solution:
Consider two circles which intersect at two points A and B
We have to prove that two circles cannot intersect at more than two points.
If the circle intersect at three points say A, B and C
Clearly, A, B and C are not collinear.
We know that through three non collinear points A, B and C only one circle can pass
This implies that two circles cannot pass through three points
Therefore, two circles cannot intersect at more than two points.
✦ Try This: ABCD is a cyclic quadrilateral in which ∠DAC = 27°; ∠DBA = 50° and ∠ADB = 33°. Calculate ∠DCB
Given, ABCD is a cyclic quadrilateral
∠DAC = 27°; ∠DBA = 50° and ∠ADB = 33°
∠ACB = ∠ABD = 33°
∠ACD = ∠ABD = 50°
We know that the angles subtended by the same chord on the circle are equal.
So, ∠DCB = ∠ACD + ∠ACB
∠DCB = 50° + 33°
Therefore, ∠DCB = 83°
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.4 Sample Problem 1
Prove that two circles cannot intersect at more than two points
Summary:
A circle can be defined as a 2D figure formed by a set of points that are adjacent to each other and are equidistant from a fixed point. It is proven that two circles cannot intersect at more than two points
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