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Standard XII

Mathematics

Auxiliary Circle of Ellipse

Any tangent t...

Question

Any tangent to the circle $x^{2} + y^{2} = a^{2}$ meets the axes of co-ordinates in $A$ and $B$ respectively. The rectangle $O A C B$ is completed. The locus of the vertex of the rectangle $O A C B$ is the curve $x^{- 2} + y^{- 2} = a^{- 2}$ .

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Solution

Let tangent $A B$ be $\frac{x}{α} + \frac{y}{β} = 1$ .
$p = r$ gives $\frac{1}{α^{2}} + \frac{1}{β^{2}} = \frac{1}{a^{2}}$
$∴$ Locus of vertex $C (α, β)$ is
$x^{- 2} + y^{- 2} = a^{- 2}$ .

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Any tangent to the circle x2+y2=a2 meets the axes of co-ordinates in A and B respectively. The rectangle OACB is completed. The locus of the vertex of the rectangle OACB is the curve x−2+y−2=a−2.

Let tangent AB be xα+yβ=1.p=r gives 1α2+1β2=1a2∴ Locus of vertex C(α,β) isx−2+y−2=a−2.

Any tangent to the circle $x^{2} + y^{2} = a^{2}$ meets the axes of co-ordinates in $A$ and $B$ respectively. The rectangle $O A C B$ is completed. The locus of the vertex of the rectangle $O A C B$ is the curve $x^{- 2} + y^{- 2} = a^{- 2}$ .

Let tangent $A B$ be $\frac{x}{α} + \frac{y}{β} = 1$ .
$p = r$ gives $\frac{1}{α^{2}} + \frac{1}{β^{2}} = \frac{1}{a^{2}}$
$∴$ Locus of vertex $C (α, β)$ is
$x^{- 2} + y^{- 2} = a^{- 2}$ .