JEE Advanced 2021 · Paper 1, Mathematics question 1

Vertices. One vertex is where the two given sides meet: $A=(-1,0)$. Let $B$ be the other vertex on the $x$-axis and $C$ the other vertex on $x+y+1=0$.

• The altitude from $C$ is perpendicular to the $x$-axis and passes through $H=(1,1)$, so it is $x=1$. Then $C=(1,-2)$.
• The altitude from $B$ is perpendicular to $x+y+1=0$ and passes through $H$, so it is $y=x$. Then $B=(0,0)$.

Circle. Since it passes through the origin, write it as $x^2+y^2+2gx+2fy=0$.

• At $(-1,0)$: $1-2g=0$, so $g=\frac12$.
• At $(1,-2)$: $5+1-4f=0$, so $f=\frac32$.

The circle is $x^2+y^2+x+3y=0$. (B)