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

Given $O(0,0)$ is the orthocentre.
Let $A(h,k)$ be 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 lines through $B$ and $C$ has the slope $\frac{-4}{7}$, By the property of the orthocentre, these two lines must be perpendicular, so we have
$\frac{k}{h}\times \frac{-4}{7}=-1$ we get $k=\frac{7}{4}\times h......\left(i\right)$ [usingthe concept that if line $L_1$ and $L_2$ are perpendicular to each other then $m_1\times m_2=-1$ ]
Also, $\frac{5-2+h}{3}+\frac{-1+3+k}{3}=7$ [using centroid formula ]
$h+k=16$ .......... (ii)
Using (i) in (ii) we get $\frac{11}{4}\times h=16$ $\therefore h=\frac{64}{11}$
And thus $k=\frac{112}{11}$
None of the options among A,B and C satisfy . Hence D is the correct option.