question_answer

Two vertices of a triangle are \[(5,-1)\] and \[(-\,2,3)\] if orthocenter of the triangle is origin, then the coordinate of third vertex is

A) \[(4,7)\]             

B) \[(3,7)\]

C) \[(-4,-7)\]           

D) \[(4,-\,7)\]

Correct Answer: C

Solution :

[c]

Slope of \[AH=\frac{k}{h}\]

Slope of\[BC=\frac{4}{-\,7}\]

Slope of \[AH\times \]slope of \[BC=-1\]

\[\therefore \]      \[\frac{k}{h}\times \frac{4}{-\,7}=-1\]

\[\frac{k}{h}=\frac{7}{4}\]                                                        ? (i)

Slope of \[AC=\frac{k+1}{h-5}\]

Slope of \[BH=\frac{-\,3}{2}\]

Slope of \[AC\times \]Slope of \[BH=-\,1\]

\[\therefore \]      \[\left( \frac{k+1}{h-5} \right)\left( \frac{-\,3}{2} \right)=-1\]

\[\Rightarrow \]   \[\frac{k+1}{h-5}=\frac{2}{3}\]                         ? (ii)

On solving Eqs. (i) and (ii), we get \[h=-\,4,k=-\,7\]

\[\therefore \]Third vertex of triangle is \[(-\,4,-7).\]