Question

If two vertices of a triangle are $(-2,3)$ and $(5,-1)$, orthocentre lies at the origin and centroid on the line $x+y=7,$ then the third vertex lies at

• A. $(7,4)$
• B. $(8,14)$
• C. $(12,21)$
• D. none of these

Answer 1

The correct option is D none of these

Let $0(0,0)$ be the orthocentre; $A(h,k)$ the third vertex;

$B(-2,3)$ and $C(5,-1)$ the other two vertices.
Then the slope of the line through $A$ and $O$ is $\frac{k}{h}$,

while the line through $B$ and $C$ has the slope $\frac{(-1-3)}{(5+2)}=-\frac{4}{7}.$

By the property of the orthocentre, these two lines must be perpendicular, so we have

$\left(\frac{k}{h}\right)\times (-\frac{4}{7})=-1\Rightarrow \frac{k}{h}=\frac{7}{4}$ ....(i)
Also $\frac{5-2+h}{3}+\frac{-1+3+k}{3}=7\Rightarrow h+k=16$ ....(ii)

which is not satisfied by the points given in $\left(a\right),\left(b\right)$ or $\left(c\right)$