Assertion -There are exactly two complex numbers which satisfy the complex equations -z - 4 - 5i- 4 and Arg z - 3- 4i -4 simultaneously- Reason- A line cuts the circle in atmost two points-

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Geometrical Representation of Argument and Modulus

Assertion :Th...

Question

Assertion :There are exactly two complex numbers which satisfy the complex equations $| z - 4 - 5 i | = 4$ and $A r g (z - 3 - 4 i) = \frac{π}{4}$ simultaneously. Reason: A line cuts the circle in atmost two points.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is D Assertion is incorrect but Reason is correct
$| z - 4 - 5 i | = 4$ represents a circle with centre (4, 5) and radius 4.
$a r g (z - 3 - 4 i) = \frac{π}{4}$ represents a ray emanating from point $(3, 4)$
Ray will intersect the circle at only one point.
So statement (I) is false and statement (II) is true.

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Assertion :There are exactly two complex numbers which satisfy the complex equations |z−4−5i|=4 and Arg(z−3−4i)=π4 simultaneously. Reason: A line cuts the circle in atmost two points.

The correct option is D Assertion is incorrect but Reason is correct|z−4−5i|=4 represents a circle with centre (4, 5) and radius 4.arg(z−3−4i)=π4 represents a ray emanating from point (3,4)Ray will intersect the circle at only one point.So statement (I) is false and statement (II) is true.

Assertion :There are exactly two complex numbers which satisfy the complex equations $| z - 4 - 5 i | = 4$ and $A r g (z - 3 - 4 i) = \frac{π}{4}$ simultaneously. Reason: A line cuts the circle in atmost two points.