# a variable straight AB intersecting the x and y axes at A & B always passes through a fixed point(a,b).Find the locus of the point dividing AB in 2:1

jagdish singh singh
173 Points
5 years ago
$\hspace{-0.7 cm}Let equation of variable line AB be y-b=m(x-a)\\\\ Now if line intersect x- axis , Then put y=0, We get \displaystyle A\left(a-\frac{b}{m},0\right)\\\\ Now if line intersect y- axis, Then put x=0, We get \displaystyle B\left(0,b-am\right)\\\\ Now using Section formula for line AB\\\\ Let M(h,k) be the point whose locus is to be calculated, Then\\\\$
jagdish singh singh
173 Points
5 years ago
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$\hspace{-0.7 cm}h=\frac{a-\frac{b}{m}}{3} \Rightarrow m=\frac{b}{a-3h} and k = \frac{2(b-am)}{3}\Rightarrow m=\frac{2b-3k}{2a}\\\\ So after eliminating m, We get \frac{b}{a-3h} = \frac{2b-3k}{a}\Rightarrow 6bh+3ak=ab \\\\ So 6bx+3ay=ab\Rightarrow \frac{6x}{a}+\frac{3y}{b}=1$