Circles
Circle
Allen Star Batch
Grade 11
Question:
Six points $(x_i, y_i); i = 1,2,3,4,5,6$ are taken on the circle $x^2 + y^2 = 4$ such that $\sum_{i=1}^{6} x_i = 8$ and $\sum_{i=1}^{6} y_i = 4$. The line segment joining orthocenter of a triangle made by any three points and the centroid of the triangle made by other three points passes through a fixed point $(h, k)$. The value of $h + k$ is ______.
Step-by-Step Solution
Key Concept: The line connecting the circumcenter and centroid of a triangle passes through a fixed point independent of the triangle's configuration.
The circumcenter $H$ of three points $A_1, A_2, A_3$ lies at the origin, while the centroid $G$ is at $\left(\frac{8-a}{3}, \frac{4-b}{3}\right)$. The line joining $H$ and $G$ satisfies the equation $(2-a)(y-b) = (1-b)(x-a)$. Substituting $x = 2$ and $y = 1$: $(2-a)(1-b) = (1-b)(2-a)$, which is always true. Thus the line $HG$ always passes through the fixed point $(2, 1)$ regardless of the values of $a$ and $b$.
Correct Answer: 3