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Question
Assertion :The common chord of the circles x2+y2−10x+16=0 and x2+y2=r2 is of maximum length if r2=34 Reason: The common chord of two circles is of maximum length if it passes through the centre of the circle with smaller radius.
Assertion :The common chord of the circles x2+y2−10x+16=0 and x2+y2=r2 is of maximum length if r2=34 Reason: The common chord of two circles is of maximum length if it passes through the centre of the circle with smaller radius.
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Solution
The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Consider the circle
S1:x2+y2−10x+16=0
(x−5)2+y2−25+16=0
(x−5)2+y2=32
Center is (5,0) and radius is 3.
Now the circle S2:x2+y2=r2 cuts the above circle, and will have the common chord of maximum length if the chord passes through the center of the other circle.
Equation of the common chord will be
S2−S1=0
−10x+16=−r2
r2=10x−16 ........... (1)
At C=(5,0)
r2=50−16 .......... (Since, (5,0) lies on (1)) =34
∴ r2=34
Hence, both assertion and reason are correct.
Consider the circle
S1:x2+y2−10x+16=0
(x−5)2+y2−25+16=0
(x−5)2+y2=32
Center is (5,0) and radius is 3.
Now the circle S2:x2+y2=r2 cuts the above circle, and will have the common chord of maximum length if the chord passes through the center of the other circle.
Equation of the common chord will be
S2−S1=0
−10x+16=−r2
r2=10x−16 ........... (1)
At C=(5,0)
r2=50−16 .......... (Since, (5,0) lies on (1))
=34
∴ r2=34
∴ r2=34
Hence, both assertion and reason are correct.
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