Question

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

• A. 7
• B. -1
• C. -19/8
• D. none

Answer 1

Answer: (A)

7

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

When centroid, orthocentre, incentre

A circumcentre lie on same straight

line, then Triangle is isoceles triangle.

$\therefore AB=AC$

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

on Squaring Both sides

$(3{)}^2+(4{)}^2=(-3{)}^2+9+k^2-6k$

$16=9+k^2-6k$

$k^2-6k-7=0$

$k^2-7k+k-7=0$

$k(k-7)+1(k-7)=0$

$(k-7)(k+1)=0$

$\therefore k=7$ or $k=-1$