If the orthocentre of the triangle formed by the lines 2 x+3 y-1=0, x+2 y-1=0 and a x+b y-…

Question

If the orthocentre of the triangle formed by the lines $2 x+3 y-1=0, x+2 y-1=0$ and $a x+b y-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$

Accepted Answer

Let circumcentre, orthocentre and centroid of a triangle $PQR$ are ${C}_{1},{H}_{1}$ and ${G}_{1}$ respectively

$\Rightarrow {G}_{1}\equiv (0,0)$ orthocentre of $	riangleABC$ is $(0,0)$

$\begin{gathered} {m}_{A{H}_{2}}=+\frac{b}{a}\Rightarrow a+b=0 \ {}^{}{H}_{2}Cy=\frac{3}{2}x \ \Rightarrow C\equiv (\frac{1}{4},\frac{3}{8})ax+by-1=0 \ \Rightarrow \frac{a}{4}+\frac{3}{8}b-1=0\Rightarrow \frac{a}{4}-\frac{3}{8}a-1=0 \ \Rightarrow a=-8,b=8 \ |a-b|=16 \end{gathered}$

Solutions · 1