A point $M$ divides $A$ and $B$ in the ratio $1:2$ where $A$ and $B$ diametrically opposite ends of a circle $x^2 + y^2 - 5x - 9y + 22 = 0$ square $AMCD$ and $BMEF$ on the length $AM$ and $MB$ are constructed on the same side of line $AB$ if co-ordinates of $A$ is $(1, 3)$ then find the orthocentre of $\triangle ABE$.
Step-by-Step Solution
Key Concept: The orthocenter of a triangle can be identified as the intersection point of altitudes, which in rectangle configurations provides symmetric geometric relationships.
From the figure, point $C(1,5)$ is the orthocenter of triangle $ABE$ where $A(1,3)$, $B(4,6)$, and $E(0,5)$. Similarly, for another triangle configuration, the orthocenter is located at $C(3,3)$. These special points arise as the intersection of altitudes in the respective triangles formed by vertices of the rectangle.
Correct Answer: 2,3