Question

Three vertices of a triangle are $A(4,3);B(1,-1)$ and $C(7,k)$. Value$\left(s\right)$ of $k$ for which centroid, orthocentre, incentre and circumcentre of the $△ABC$ lie on the same straight line is/are :

• A. $7$
• B. $-1$
• C. $-19/8$
• D. $none$

Answer 1

The correct option is A $7$

Vertices are $A(4,3),B(1,-1),C(7,k)$

centroid, arthocenter, incenter & circumcenter

of $△ABC$ lie on same line so $△ABC$ is

isosceles $△.$

$\therefore AB=AC$

$A{B}^2=A{C}^2$

$\sqrt{(1-4{)}^2+(-1-3{)}^2}=\sqrt{(7-4{)}^2+(k-3{)}^2}$

$=9+16=9+(k-3{)}^2$

$=16(k-3{)}^2$

$\therefore k-3=4$ or $k-3=-4$

$\Rightarrow k=7$ $\Rightarrow k=-1$