JEE Advanced 2021 Paper 1 · Q01 · Circles

Question

Consider a triangle $\Delta$ whose two sides lie on the x-axis and the line $x+y+1=0$. If the orthocentre of $\Delta$ is $(1, 1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is

SolveKar AI · Accepted Answer

The two sides lie on the x-axis ($y=0$) and the line $x+y+1=0$. These intersect at vertex $A=(-1,0)$. The image of the orthocentre $H=(1,1)$ in any side of the triangle lies on the circumcircle. Reflecting $H=(1,1)$ across the x-axis gives $(1,-1)$, which must lie on the circumcircle. Reflecting $H$ across the line $x+y+1=0$ also yields a point on the circumcircle. Checking each option: option (B) $x^2+y^2+x+3y=0$ is satisfied by both $(-1,0)$: $1+0-1+0=0$ \checkmark and $(1,-1)$: $1+1+1-3=0$ \checkmark. Hence (B) is the correct equation of the circumcircle.

JEE Advanced 2021 Paper 1 · Q01 · Circles

Solution