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Section Formula Using Complex Numbers

Let A, B, C...

Question

Let $A, B, C$ represent the complex number $z_{1}, z_{2}, z_{3}$ respectively on the complex plane. If the circumcentre of the triangle $A B C$ lies at the origin, then the orthocentre is represented by the complex number.

z_{1} + z_{2} - z_{3}

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z_{2} + z_{3} - z_{1}

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z_{3} + z_{1} - z_{2}

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z_{1} + z_{2} + z_{3}

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Solution

The correct option is D $z_{1} + z_{2} + z_{3}$
Centre of the circle is ato origin $= 0$
Centroid $= \frac{z_{1} + z_{2} + z_{3}}{3}$
Refer image
From orthocentre, $O C = 3 P C = \frac{3 (z_{1} + z_{2} + z_{3})}{3}$
$= z_{1} + z_{2} + z_{3}$
The orthocentre complex no. is $z_{1} + z_{2} + z_{3}$

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| z_{1} + z_{2} + z_{3} |

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