If the circumcenter of an acute-angled triangle lies at the origin and the centroid is the middle point of the line joining the points `(a^2+1,a^2+1)` and `(2a ,-2a),` then find the orthocentre.

Options

A. `y=(a^(2)+1)x`
B. `y=2ax`
C. `x=y=0`
D. `(a-1)^(2)x-(a+1)^(2)y=0`

Answer

D

Explanation

The circumcentre is at the origin and the centroid is at `((a^(2)+2a)/(2),(a^(2)+1-2a)/(2))i.e. at (((a+1)^(2))/(2),((a-1)^(2))/(2))`
Let (x,y) be the coordinates of the orthocentre. Sice centroid divides the line segment joining circumentre and orthocentre in the ratio `1:2`,
`:. ((a+1)^(2))/(2)=(x)/(3)and (a(a-1)^(2))/(2)=(y)/(3)`
`rArr((a+1)^(2))/((a-1)^(2))=(x)/(y)rArr (a-1)^(2)x=(a+1)^(2)y`