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Question

The altitude from the vertices A,B and C of the triangle ABC meet its circumcircle at D,E and F respectively. The complex numbers representing the points D,E and F are z1,z2 and z3 respectively. If z3z1z2z1 is purely real, then the triangle ABC is right angle at

A
A
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B
B
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C
C
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D
None of these
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Solution

The correct option is A A

FDA=FCA=90A
ADE=ABE=90A
FDE=1802A=π2A
Similarly DFE=π2C and DEF=π2B
The angles of ΔDEF are π2A,π2B and π2C, respectively.
Also it is given that z3z1z2z1 is purely real. Hence
arg(z3z1z2z1)=0 or π
π2A=0 or π
A=π2 or 0 (not permissible)
Hence triangle ABC is right angled at A.

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