The vertices of a triangle are at $-\hat{i}+3 \hat{j}$ and $2 \hat{i}+5 \hat{j}$ and its orthocenter is at $\bar{i}+2 \bar{j}$. If the position vector of the third vertex is $a \hat{i}+b \hat{j}$, then $(a, b)=$

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Aicharya · Accepted Answer

Let $A=(-1,3), B=(2,5), C=(a, b), H=(1,2)$.

$\begin{aligned} & A E \perp B C \\ & M_{A E} \times M_{B C}=-1\end{aligned}$ $\Rightarrow \quad\left(\frac{3-2}{-1-1}\right)\left(\frac{5-b}{2-a}\right)=-1$ $\Rightarrow \quad 2 a-b=-1$ ...(i) Again $m_{A B} \times m_{C D}=-1$ $\Rightarrow\left(\frac{5-3}{2+1}\right)\left(\frac{b-2}{a-1}\right)=-1$ $\Rightarrow \quad 3 a+2 b=7$ ...(ii) Multiplying of 2 in Eq. (i) and adding Eqs. (i) and (ii)

From Eq. (i), $b=2 a+1=2 \times \frac{5}{7}+1=\frac{17}{7}$ $\therefore \quad(a, b)=\left(\frac{5}{7}, \frac{17}{7}\right)$

The vertices of a triangle are at $-\hat{i}+3 \hat{j}$ and $2 \hat{i}+5 \hat{j}$ and its orthocenter is at…

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