If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1)

Question

If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$

Accepted Answer

Equation of CE

$\begin{gathered} y-1=-(x-3) \ x+y=4 \end{gathered}$

orthocentre lies on the line $x+y=4$

so, $a+b=4$

${I}_{1}=\int _{a}^{b}x\sin (x(4-x))dx\,$ ..... (i)

Using king rule

${I}_{1}=\int _{a}^{b}(4-x)\sin (x(4-x))dx\,$ .... (ii)

$\begin{gathered} + \ 2{I}_{1}=\int _{a}^{b}4\sin (x(4-x))dx \ 2{I}_{1}=4{I}_{2} \ {I}_{1}=2{I}_{2} \ \frac{{I}_{1}}{{I}_{2}}=2 \ \frac{36{I}_{1}}{{I}_{2}}=72 \end{gathered}$

Solutions · 1