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Grade 11Analytical Geometry

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

Profile image of Khalid yahiya Ahmed
9 Years agoGrade 11
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2 Answers

Profile image of jagdish singh singh
9 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\\\\
Profile image of jagdish singh singh
9 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$

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