Question

Two vertices of a triangle are $(3,-2)$ and $(-2,3)$ and its orthocentre is $(-6,1)$. Then the third vertex or this triangle can NOT lie on the line:

• A. $6x+y=0$
• B. $4x+y=2$
• C. $5x+y=2$
• D. $3x+y=3$

Answer 1

The correct option is A $6x+y=0$

$\begin{array}{c}\text{Ginen wertices of}△\text{are}(3,-2)\text{and}(-2,3)\\ \text{orthocenter of}△\text{is}(-6,1)\\ \text{To find the third vertice}-(h,k)\\ \left(\frac{k-1}{n+6}\right)\times \left(\frac{3+2}{-2-3}\right)=-1\\ \left(\frac{1+2}{-6-3}\right)\times \left(\frac{k-3}{n+2}\right)=-1\end{array}$

$\begin{array}{cc}\text{on solving (1)}and\left(2\right)weget& \\ & h-k+7=0,3h-k+9=0\\ \Rightarrow & h=-1,k=-6\end{array}$

$\begin{array}{c}\text{After analysing the options only option}A\text{)}\\ \text{i.e}6x+y=0\text{is not satisfied by}(h,k)\\ \text{Hence,}\left(A\right)\text{is the correct option.}\end{array}$