Assertion -If z is a complex number such that -2z-3-3z-1- - -5-3- then z lies on a circle- Reason- If a- b are two complex numbers k - 0- k-1- then -z-a-z-b- - k represents a circle-

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Standard XII

Mathematics

Arithmetic Progression

Assertion :If...

Question

Assertion :If $z$ is a complex number such that $∣ ∣ ∣ \frac{2 z + 3}{3 z + 1} ∣ ∣ ∣ = \sqrt{\frac{5}{3}}$ , then $z$ lies on a circle. Reason: If $a, b$ are two complex numbers $k > 0, k \neq 1$ , then
$∣ ∣ ∣ \frac{z + a}{z + b} ∣ ∣ ∣ = k$ represents a circle.

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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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Simplifying
$| \frac{z + \frac{3}{2}}{z + \frac{1}{3}} | = \frac{3}{2} \sqrt{\frac{5}{3}}$
Hence
$| \frac{z + \frac{3}{2}}{z + \frac{1}{3}} | = C$ represents a circle since
$C > 0$ and $C \neq 1$
Hence
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
Hence, option 'A' is correct.

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Assertion :If z is a complex number such that ∣∣∣2z+33z+1∣∣∣=√53, then z lies on a circle. Reason: If a,b are two complex numbers k>0,k≠1, then∣∣∣z+az+b∣∣∣=k represents a circle.

Assertion :If $z$ is a complex number such that $∣ ∣ ∣ \frac{2 z + 3}{3 z + 1} ∣ ∣ ∣ = \sqrt{\frac{5}{3}}$ , then $z$ lies on a circle. Reason: If $a, b$ are two complex numbers $k > 0, k \neq 1$ , then
$∣ ∣ ∣ \frac{z + a}{z + b} ∣ ∣ ∣ = k$ represents a circle.