If two vertices of a triangle are `(-2,3)` and `(5,-1)` the orthocentre lies at the origin, and the centroid on the line `x+y=7` , then the third vertex lies at `(7,4)` (b) `8,14)` `(12 ,21)` (d) none of these

Options

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

Answer

D

Explanation

Given O(0,0) is the orhtocenter . Let A(h,k) be the third vertex, and `B(-2,3)` and `C(5,-1)` the other two vertices. Then the slope of the line through A and O is k/h. while the line through B and C has the slope `-4//7`. By the property of the orthocenter, these two lines must be perpendiuclar. So, we have `(k/(h))(-(4)/7)=-1or (k)/(h)=(7)/(4)`....(1)
Also, ` (5-2+h)/(3)+(-1+3+k)/(3)=7` ....(ii)
`h+k=16`