Question

$ABC$ is a variable triangle such that $A$ is $(1,2)$, $B$ and $C$ lie on line $y=x+\lambda$ (where $\lambda$ is a variable), then locus of the orthocentre of $△ABC$ is

• A. $(x-1{)}^2+y^2=4$
• B. $x+y=3$
• C. $2x-y=0$
• D. None of these

Answer 1

Answer: (B)

$x+y=3$

Altitude $AL$ will be perpendicular to $BC$ thus equation of $AL$ will be $y+x=c$, where $c$ is some constant

[using the concept that if line $L_1$ and $L_2$ are at right angles to each other then $m_1\times m_2=-1$ where $m_1$ and $m_2$ are slopes of $L_1$ and $L_2$ respectively]

Since $AL$ passes through $A$, then $c=2+1=3$

Therefore line $AL$ is $y+x=3$ orthocentre lies on $AL$. Hence locus of orthocentre is $y+x=3$