Question

A(1, 0) and B(0, 1) are two fixed points 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

Answer: (A)

x^2 + y^2 - 2x - 2y + 1 = 0

Answer 2

The circle $x^2+y^2=1$ is the circumcircle of $\triangle ABC$ with center O(0,0). The orthocentre H(h,k) of $\triangle ABC$ is related to the vertices A, B, C and the circumcenter O by the vector equation $\vec{OH} = \vec{OA} + \vec{OB} + \vec{OC}$. Substituting the coordinates A(1,0), B(0,1), C($x_C, y_C$) and H(h,k), we get $(h,k) = (1+x_C, 1+y_C)$. Since C is on the circle, $x_C^2+y_C^2=1$. Substituting $x_C=h-1$ and $y_C=k-1$ into this equation gives $(h-1)^2+(k-1)^2=1$, which simplifies to $h^2+k^2-2h-2k+1=0$. The locus is $x^2+y^2-2x-2y+1=0$.