A variable line is at a constant distance p from the origin and meets coordinate axes in A and B. Show that the locus of the centroid of the $Δ$ OAB is $x^{- 2} + y^{- 2} = p^{- 2}$ .

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Solution

Area of $Δ$ AOB
$\frac{1}{α} \times (A O \times B O) = \frac{1}{2} (A B) (O C)$
$\frac{1}{2} (x) (y) = \frac{1}{2} P \sqrt{x^{2} + y^{2}}$
$\frac{x y}{\sqrt{x^{2} + y^{2}}} = P$
$\Rightarrow \frac{1}{P} = \frac{\sqrt{x^{2} + y^{2}}}{x y}$
$\Rightarrow \frac{1}{P^{2}} = \frac{x^{2} + y^{2}}{x^{2} y^{2}}$
$\Rightarrow \frac{1}{P^{2}} = \frac{1}{x^{2}} + \frac{1}{y^{2}}$
$\Rightarrow P^{- 2} = x^{- 2} + y^{- 2}$ .

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A variable line is at a constant distance p from the origin and meets coordinate axes in A and B. Show that the locus of the centroid of the ΔOAB is x−2+y−2=p−2.

Area of ΔAOB1α×(AO×BO)=12(AB)(OC)12(x)(y)=12P√x2+y2xy√x2+y2=P⇒1P=√x2+y2xy⇒1P2=x2+y2x2y2⇒1P2=1x2+1y2⇒P−2=x−2+y−2.

A variable line is at a constant distance p from the origin and meets coordinate axes in A and B. Show that the locus of the centroid of the $Δ$ OAB is $x^{- 2} + y^{- 2} = p^{- 2}$ .