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  •  If two vertices of a triangle are and <img alt="(5,-1)" src="https://entrancecorner.oncodecogs.com/gif.

 If two vertices of a triangle are (-2,3) and (5,-1), orthocentre lies at the origin and centroid on the line \mathrm{x+y=7}, then the third vertex lies at

Option: 1

(7,4)


Option: 2

(8,14)


Option: 3

(12,21)


Option: 4

none of these


Answers (1)

best_answer

Let \mathrm{O(0,0)}be the orthocentre; \mathrm{A(h, k)} the third vertex; \mathrm{B(-2,3)} and \mathrm{C(5,-1)} the other two vertices.
Then the slope of the line through \mathrm{A} and \mathrm{O} is \mathrm{\frac{k}{h}}, while the line through \mathrm{B} and \mathrm{C} has the slope

\mathrm{\frac{(-1-3)}{(5+2)}=-\frac{4}{7}}
\mathrm{We \, have, \left(\frac{k}{h}\right)\left(-\frac{4}{7}\right)=-1 \Rightarrow \frac{k}{h}=\frac{7}{4}\quad \cdots(i)}
\mathrm{Also \quad \frac{5-2+h}{3}+\frac{-1+3+k}{3}=7 \quad \Rightarrow h+k=16}\quad \cdots(ii)

,Solving equations (i) and (ii) \mathrm{h=\frac{64}{11}} and \mathrm{k=\frac{112}{11}}

Posted by

Sanket Gandhi

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