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