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Q.

Consider the circle $| z - 5 i | = 4. z_{1} a n d z_{2}$ are two points on it such that $| z_{2} | > | z_{1} | a n d A r g (z_{1}) = A r g (z_{2}) = \frac{3 π}{10} .$ A tangent to circle at $z_{2}$ intersect real axis at $z_{3},$ then $\arg (\frac{5 i - z_{3}}{z_{2} - z_{3}}) =$

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a

$\frac{π}{5}$

b

not possible to determine

c

$\frac{3 π}{10}$

d

$\frac{2 π}{5}$

answer is C.

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Detailed Solution

detailed_solution_thumbnail

Clearly the points OBC and D are concyclic

$H e n c e ∠ D O C = ∠ D B C = \frac{2 π}{10} = \frac{π}{5}$

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Consider the circle |z−5i|=4. z1 and z2 are two points on it such that |z2|>|z1| and Arg(z1)=Arg(z2)=3π10. A tangent to circle at z2 intersect real axis at z3, then arg(5i−z3z2−z3)=