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The-equation-of-an-ellipse-is-given-as-x-2-y-2xcot2-2-1-0-lt-lt-0-25pi-Show-that-the-minimum-and-maximum-distances-from-the-centre-to-the-circumference-of-this-ellipse-are-tan-and-cot

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Question Number 2168 by Yozzi last updated on 06/Nov/15
The equation of an ellipse is given as x^2 +(y+2xcot2θ)^2 =1 (0<θ<0.25π). Show that the minimum and maximum distances from the centre to the circumference of this ellipse are tanθ and cotθ respectively.
$${The}\:{equation}\:{of}\:{an}\:{ellipse}\:{is}\:{given} \\ $$$${as}\: \\ $$$$\:\:{x}^{\mathrm{2}} +\left({y}+\mathrm{2}{xcot}\mathrm{2}\theta\right)^{\mathrm{2}} =\mathrm{1}\:\:\:\:\left(\mathrm{0}<\theta<\mathrm{0}.\mathrm{25}\pi\right). \\ $$$${Show}\:{that}\:{the}\:{minimum}\:{and}\:{maximum} \\ $$$${distances}\:{from}\:{the}\:{centre}\:{to}\:{the}\: \\ $$$${circumference}\:{of}\:{this}\:{ellipse}\:{are} \\ $$$${tan}\theta\:{and}\:{cot}\theta\:{respectively}.\: \\ $$$$ \\ $$
Answered by prakash jain last updated on 07/Nov/15
x^2 +y^2 +4xycot 2θ+4x^2 cot^2 2θ−1=0 x^2 (1+4cot^2 2θ)+y^2 +4xycot 2θ−1=0 x=x_1 cos α+y_1 sin α y=−x_1 sin α+y_1 cos α xy co−efficient after rotation 2(1+4cot^2 2θ)cos αsin α−2sin αcos α+4cos2αcot2θ=0 4cot^2 2θsin 2α+4cos2αcot 2θ=0 α=−θ New coefficient after rotation ax^2 +bxy+cy^2 +dx+ey+f=0 a=(1+4cot^2 2θ)cos^2 θ−4cot2θsin θcosθ+sin^2 θ b=0 c=(1+4cot^2 2θ)sin^2 θ+4cot2θsin θcos θ+cos^2 θ d=0 e=0 f=1 more simplication to be done but it should reduce to the form (x_1 ^2 /A^2 )+(y_1 ^2 /B^2 )=1
$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{4}{xy}\mathrm{cot}\:\mathrm{2}\theta+\mathrm{4}{x}^{\mathrm{2}} \mathrm{cot}^{\mathrm{2}} \mathrm{2}\theta−\mathrm{1}=\mathrm{0} \\ $$$${x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{4cot}^{\mathrm{2}} \mathrm{2}\theta\right)+{y}^{\mathrm{2}} +\mathrm{4}{xy}\mathrm{cot}\:\mathrm{2}\theta−\mathrm{1}=\mathrm{0} \\ $$$${x}={x}_{\mathrm{1}} \mathrm{cos}\:\alpha+{y}_{\mathrm{1}} \mathrm{sin}\:\alpha \\ $$$${y}=−{x}_{\mathrm{1}} \mathrm{sin}\:\:\alpha+{y}_{\mathrm{1}} \mathrm{cos}\:\alpha \\ $$$${xy}\:{co}−{efficient}\:{after}\:{rotation} \\ $$$$\mathrm{2}\left(\mathrm{1}+\mathrm{4cot}^{\mathrm{2}} \mathrm{2}\theta\right)\mathrm{cos}\:\alpha\mathrm{sin}\:\alpha−\mathrm{2sin}\:\alpha\mathrm{cos}\:\alpha+\mathrm{4cos2}\alpha\mathrm{cot2}\theta=\mathrm{0} \\ $$$$\mathrm{4cot}^{\mathrm{2}} \mathrm{2}\theta\mathrm{sin}\:\mathrm{2}\alpha+\mathrm{4cos2}\alpha\mathrm{cot}\:\mathrm{2}\theta=\mathrm{0} \\ $$$$\alpha=−\theta \\ $$$$\mathrm{New}\:\mathrm{coefficient}\:\mathrm{after}\:\mathrm{rotation} \\ $$$${ax}^{\mathrm{2}} +{bxy}+{cy}^{\mathrm{2}} +{dx}+{ey}+{f}=\mathrm{0} \\ $$$${a}=\left(\mathrm{1}+\mathrm{4cot}^{\mathrm{2}} \mathrm{2}\theta\right)\mathrm{cos}^{\mathrm{2}} \theta−\mathrm{4cot2}\theta\mathrm{sin}\:\theta\mathrm{cos}\theta+\mathrm{sin}^{\mathrm{2}} \theta \\ $$$${b}=\mathrm{0} \\ $$$${c}=\left(\mathrm{1}+\mathrm{4cot}^{\mathrm{2}} \mathrm{2}\theta\right)\mathrm{sin}^{\mathrm{2}} \theta+\mathrm{4cot2}\theta\mathrm{sin}\:\theta\mathrm{cos}\:\theta+\mathrm{cos}^{\mathrm{2}} \theta \\ $$$${d}=\mathrm{0} \\ $$$${e}=\mathrm{0} \\ $$$${f}=\mathrm{1} \\ $$$${more}\:{simplication}\:{to}\:{be}\:{done}\:{but}\:{it}\:{should} \\ $$$${reduce}\:{to}\:{the}\:{form} \\ $$$$\frac{{x}_{\mathrm{1}} ^{\mathrm{2}} }{{A}^{\mathrm{2}} }+\frac{{y}_{\mathrm{1}} ^{\mathrm{2}} }{{B}^{\mathrm{2}} }=\mathrm{1} \\ $$$$ \\ $$

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