Ex - 10.3
Question 1. Draw different pair of circles. How many points does each pair have in common? What is the maximum number of common points?
Solution
Question 1. Draw different pair of circles. How many points does each pair have in common? What is the maximum number of common points?
Solution
Consider the following pair of circles.
(i) circles don't intersect each other at any point, so circles are not having any point in common.
(i) circles don't intersect each other at any point, so circles are not having any point in common.
(ii) Circles touch each other only at one point P so there is only 1 point in common.
(iii) Circles touch each other at 1 point X only. So the circles have 1 point in common.
(iv) These circles intersect each other at two points P and Q. So the circles have two points in common. We may observe that there can be maximum 2 points in common.
(iv) These circles intersect each other at two points P and Q. So the circles have two points in common. We may observe that there can be maximum 2 points in common.
We can have a situation in which two congruent circles are superimposed on each other, this situation can be referred as if we are drawing circle two times.
Question 2. Suppose you are given a circle, Give a construction to find its centre.
Solution
Following are the steps of construction:
Step1. Take the given circle centered at point O.
Step2. Take any two different chords AB and CD of this circle and draw perpendicular bisectors of these
Step1. Take the given circle centered at point O.
Step2. Take any two different chords AB and CD of this circle and draw perpendicular bisectors of these
chords.
Step3. Let these perpendicular bisectors meet at point O. Now, O is the centre of given circle.
Step3. Let these perpendicular bisectors meet at point O. Now, O is the centre of given circle.
Question 3. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.
Solution
Consider two circles centered at point O and O' intersect each other at point A and B respectively.
Join AB. AB is the chord for circle centered at O, so perpendicular bisector of AB will pass through O.
Again AB is also chord of circle centered at O', so, perpendicular bisector of AB will also pass through O'.
Clearly centres of these circles lie on the perpendicular bisector of common chord.
Join AB. AB is the chord for circle centered at O, so perpendicular bisector of AB will pass through O.
Again AB is also chord of circle centered at O', so, perpendicular bisector of AB will also pass through O'.
Clearly centres of these circles lie on the perpendicular bisector of common chord.
1 comments:
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