Question

Point $O,A,B,C,......$ are shown in figure where $OA=2AB=4BC=.....$ so on. Let $A$ is the centroid of a triangle whose orthocentre and circumcenter are $(2,4)$ and $(\frac{7}{2},\frac{5}{2})$ respectively. If an insect starts moving from the point $O(0,0)$ along the straight line in zig-zag fashion and terminates ultimately at point $P(\alpha ,\beta )$ then find the value of $(\alpha +\beta )$.

Answer 1

Centroid divides in $2:1$ (ortho + circumcenter)

Centroid = (O) = $(\frac{2\times \frac{7}{2}+2}{3},\frac{2\times \frac{5}{2}+4}{2})$

$A=(3,3)$

OA = 2 AB = 4BC = 8CD ..... soon

So increment in +ve x coordinate

$\alpha =x=3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+.........\infty$

$\alpha =3(1+\frac{1}{2}+\frac{1}{4}.......\infty )$ Infinite GP

$\alpha =3\left(\frac{1}{1-\frac{1}{2}}\right)=6$

$y=3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\frac{3}{16}$ .......

$\downarrow$ $\downarrow$

(For (For

-ve y +ve y

direction) direction)

$y=3(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}.....\infty )$

$y=3\left(\frac{1}{1+\frac{1}{2}}\right)=2$

$\beta =2$

$P(\alpha ,\beta )=P(6,2)$

$\alpha +\beta =8$