Question Number 51817 by peter frank last updated on 30/Dec/18 $${The}\:{earth}\:{moves}\:{arround} \\ $$$${the}\:{sun}\:{in}\:{an}\:{elliptical} \\ $$$${orbit}\:{with}\:{the}\:{sun}\:{at} \\ $$$${one}\:{of}\:{its}\:{foci}.{the}\:{eccentricity} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}.{the}\:{shortest}\:{distance} \\ $$$${from}\:{the}\:{earth}\:{to}\:{sun}\: \\ $$$${is}\:\mathrm{3000000}{km}.{Find}\:{the} \\ $$$${furthest}\:{distance}\:{of}\:{the}…
Question Number 51712 by ajfour last updated on 29/Dec/18 Commented by ajfour last updated on 29/Dec/18 $${If}\:{AB}\:=\:{l}\:=\:\pi{R}/\mathrm{2}\:\:,\:{find}\:{locus}\:{of} \\ $$$${midpoint}\:{of}\:{rod}\:{as}\:{it}\:{rolls}\:{around} \\ $$$${the}\:{circle}. \\ $$ Commented by…
Question Number 182772 by Acem last updated on 14/Dec/22 $$\:\chi\:\begin{cases}{{x}=\:{t}+\mathrm{1}}\\{{y}=\:\mathrm{2}{t}−\mathrm{3}}\\{{z}=\:−{t}\:+\mathrm{2}}\end{cases}\:\:\:\Delta\begin{cases}{{x}=\:\mathrm{3}{t}\:+\mathrm{2}}\\{{y}=\:−{t}−\mathrm{1}\:\:\:}\\{{z}=\:{t}+\mathrm{1}}\end{cases}\:\begin{cases}{{Are}\:\:{these}\:{two}\:{lines}\:{located}}\\{{in}\:{the}\:{same}\:{plane}\:{or}\:{not}?}\\{{where}\:{t}\in\:\mathbb{R}}\end{cases} \\ $$$$ \\ $$ Answered by mr W last updated on 14/Dec/22 $${Method}\:{I} \\ $$$${line}\:\mathrm{1}:\:\left(\mathrm{1},−\mathrm{3},\mathrm{2}\right)+{s}\left(\mathrm{1},\mathrm{2},−\mathrm{1}\right)…
Question Number 51694 by peter frank last updated on 29/Dec/18 Answered by tanmay.chaudhury50@gmail.com last updated on 29/Dec/18 $${point}\:{p}\left({acos}\theta,{bsin}\theta\right) \\ $$$${tangent}\:{at}\:{point}\:{p}\:\:{is}\:\frac{{xcos}\theta}{{a}}+\frac{{ysin}\theta}{{b}}=\mathrm{1} \\ $$$$\frac{{ysin}\theta}{{b}}=\mathrm{1}−\frac{{xcos}\theta}{{a}} \\ $$$${y}=\frac{{b}}{{sin}\theta}−\frac{{b}}{{sin}\theta}×\frac{{cos}\theta}{{a}}{x}\:\:\left[{slope}=−\frac{{b}}{{a}}{cot}\theta\right] \\…
Question Number 51614 by peter frank last updated on 29/Dec/18 $${Prove}\:{that}\:{length}\:{of} \\ $$$${lactus}\:{rectum}\:{when}\: \\ $$$${directrix}\:{are}\:{parallel}\:{to} \\ $$$${y}−{axis}\:{is}\: \\ $$$${y}_{\mathrm{1}} −{y}_{\mathrm{2}} =\frac{\mathrm{2}{b}^{\mathrm{2}} }{{a}} \\ $$ Answered…
Question Number 51612 by peter frank last updated on 29/Dec/18 $${Find}\:{the}\:{equation}\:{of} \\ $$$${the}\:{ellipse}\:{with}\:{ecentricity} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:{and}\:{the}\:{focus}\:\left(\mathrm{2},\mathrm{1}\right) \\ $$$${Does}\:{the}\:{line}\:{x}=\mathrm{3}\:{touches} \\ $$$${ellipse}.{if}\:{so}\:{at}\:{what}\: \\ $$$${point}?{if}\:{line}\:{x}=\mathrm{5}\:{is}\:{the} \\ $$$${line}\:{of}\:{direction}. \\ $$…
Question Number 51613 by peter frank last updated on 29/Dec/18 $${Prove}\:{that}\:{the}\:{perpendicilar} \\ $$$${tangent}\:{to}\:{the}\:{ellipse} \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}\:\:{meets}\:{on}\:{the} \\ $$$${circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} .…
Question Number 51588 by peter frank last updated on 28/Dec/18 $${The}\:{tangent}\:{at}\:{P}\:\:{to}\:{an}\:{ellipse} \\ $$$${meets}\:{directrix}\:{at}\:{Q} \\ $$$${prove}\:{that}\:{the}\:{line} \\ $$$${joining}\:{the}\:{corresponding} \\ $$$${focus}\:{to}\:{P}\:{and}\:{Q}\:{are} \\ $$$${perpendicular} \\ $$ Answered by…
Question Number 51590 by peter frank last updated on 28/Dec/18 $${The}\:{line}\:{y}={mx}+{c}\:{touches} \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$${prove}\:{that}\:{the}\:{foot}\:{of}\: \\ $$$${perpendicular}\:{from} \\ $$$${focus}\:{into}\:{this}\:{line}\:{lie}\:{on} \\ $$$${auxillary}\:{circle}\:…
Question Number 51492 by peter frank last updated on 27/Dec/18 $${For}\:{ellipse}\: \\ $$$$\mathrm{16}{x}^{\mathrm{2}} +\mathrm{4}{y}^{\mathrm{2}} +\mathrm{96}{x}−\mathrm{8}{y}−\mathrm{84}=\mathrm{0} \\ $$$${find} \\ $$$$\left.{i}\right){centre} \\ $$$$\left.{ii}\right){verteces} \\ $$$$\left.{iii}\right){focus} \\ $$$$\left.{iv}\right){directrix}…