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 vert

Question

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 vert

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

Monjur Alahi

6 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - D
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`