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Question
Assertion :Let A1 and A2 represent z1 and z2.
Equation of circle with A1A2 as diameter is |2z−z1−z2|=|z1−z2| Reason: If z is any point on the circle with A1A2 as diameter, then
(z−z1)(¯z−¯z2)+(¯z−¯z1)(z−z2)=0
Assertion :Let A1 and A2 represent z1 and z2.
Equation of circle with A1A2 as diameter is |2z−z1−z2|=|z1−z2|
Reason: If z is any point on the circle with A1A2 as diameter, then(z−z1)(¯z−¯z2)+(¯z−¯z1)(z−z2)=0
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Solution
The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
The equation of the circle is (z−z1)(¯z−¯z1)+(z−z2)(¯z−¯z2)=(z1−z2)(¯z1−¯z2)=> 2z¯z−z¯z1−z1¯z−z¯z2−z2¯z+z1¯z2+¯z2z1=0 ..... (i)
The above comes from the fact that the diameter of a circle subtends a right angle at the circumference and hence, we can use Pythagoras' theorem.
In the given expression, if we multiply both sides by their conjugate we get the same thing: i.e. (2z−z1−z2)(2¯z−¯z1−¯z2)=(z1−z2)(¯z1−¯z2).
Again, using the fact that the diameter of a circle subtends a right angle at the circumference and hence:
arg(z−z2z−z1)=±π2⇒z−z2z−z1+¯z−¯z2¯z−¯z1=0⇒(z−z1)(¯z−¯z2)+(¯z−¯z1)(z−z2)=0
Hence, (a) is correct.
The equation of the circle is (z−z1)(¯z−¯z1)+(z−z2)(¯z−¯z2)=(z1−z2)(¯z1−¯z2)=> 2z¯z−z¯z1−z1¯z−z¯z2−z2¯z+z1¯z2+¯z2z1=0 ..... (i)
The above comes from the fact that the diameter of a circle subtends a right angle at the circumference and hence, we can use Pythagoras' theorem.
In the given expression, if we multiply both sides by their conjugate we get the same thing: i.e. (2z−z1−z2)(2¯z−¯z1−¯z2)=(z1−z2)(¯z1−¯z2).
Again, using the fact that the diameter of a circle subtends a right angle at the circumference and hence:
arg(z−z2z−z1)=±π2⇒z−z2z−z1+¯z−¯z2¯z−¯z1=0⇒(z−z1)(¯z−¯z2)+(¯z−¯z1)(z−z2)=0
Hence, (a) is correct.
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