Let a circle in complex plane pass through the points and . If is a point on such that the line through and is perpendicular to the line through and , then is equal to

Question

Let a circle  in complex plane pass through the points  and . If  is a point on  such that the line through  and  is perpendicular to the line through  and , then  is equal to

Canvas Classes · Accepted Answer

🧠 Three Equidistant Points ⟹ Circle is  — Then Use Slope Constraint

Read the question. Three points  all have  (check: , , ). So they lie on the circle  centred at origin. The circumcircle is this circle.

Now find  on  with line  perpendicular to line .

Slope of : from  to : slope . Perpendicular slope = .

So  on  with line from  to  having slope . Parametrise: , find  such that .

This is the habit: when three points happen to share a common modulus, the circumcircle is the obvious circle

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More questions from this chapter