Question

$A(1,0)$ and $B(0,1)$ and two fixed point on the circle $x^2+y^2=1$. C is a variable point on this circle. As C moves, the locus of the orthocentre of the triangle ABC is

• A. $x^2+y^2-2x-2y+1=0$
• B. $x^2+y^2-x-y=0$
• C. $x^2+y^2=4$
• D. $x^2+y^2+2x-2y+1=0$

Answer 1

The correct option is A $x^2+y^2-2x-2y+1=0$

Let the coordinates of third point C be $\alpha ,\beta$.
So, ${\alpha }^2+{\beta }^2=\mathrm{1....}\left(1\right)$
The coordinates of circumcenter of the triangle are $(0,0)$.
The coordinates of centroid of the triangle are $(\frac{\alpha +1}{3},\frac{\beta +1}{3})$
Centroid divides the line joining circumcenter and the orthocenter in the ratio $(1:2)$
Using the above, we get the coordinates of orthocenter as $(\alpha +1,\beta +1)$
Hence, $h=\alpha +1$ and $k=\beta +1$
Using the relation in $\left(1\right)$, we get $(h-1{)}^2+(k-1{)}^2=1$
$\Rightarrow h^2+k^2-2h-2k+1=0$