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Category: Coordinate Geometry

The-normal-at-the-point-P-4cos-3sin-on-the-ellipse-x-2-16-y-2-9-1-meets-the-x-axis-and-y-axis-at-A-and-B-respectively-show-that-locus-of-the-mid-point-of-AB-is-an-ellipse-with-the-same-e

Question Number 62753 by peter frank last updated on 24/Jun/19 $${The}\:{normal}\:{at}\:{the}\:{point} \\ $$$${P}\left(\mathrm{4cos}\:\theta,\mathrm{3sin}\:\theta\right)\:{on}\:{the} \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{\mathrm{16}}\:+\frac{{y}^{\mathrm{2}} }{\mathrm{9}}=\mathrm{1}\:{meets} \\ $$$${the}\:{x}−{axis}\:{and}\:{y}−{axis} \\ $$$${at}\:{A}\:{and}\:{B}\:{respectively} \\ $$$${show}\:{that}\:{locus}\:{of}\:{the} \\ $$$${mid}−{point}\:{of}\:{AB}\:{is}\:{an}…

Given-a-sequence-u-n-defined-reculsively-by-u-n-1-3u-n-4u-n-1-u-0-1-u-2-3-show-that-u-n-1-4u-n-1-n-1-3u-0-4u-1-hence-show-that-u-n-is-a-divegent-sequence-

Question Number 127116 by physicstutes last updated on 27/Dec/20 $$\mathrm{Given}\:\mathrm{a}\:\mathrm{sequence}\:\left({u}_{{n}} \right)\:\mathrm{defined}\:\mathrm{reculsively}\:\mathrm{by} \\ $$$$\:{u}_{{n}+\mathrm{1}} \:=\:\mathrm{3}{u}_{{n}} +\:\mathrm{4}{u}_{{n}−\mathrm{1}} ,\:\:{u}_{\mathrm{0}} =\:\mathrm{1}\:,\:{u}_{\mathrm{2}} \:=\:\mathrm{3} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:\:{u}_{{n}+\mathrm{1}} −\mathrm{4}{u}_{{n}} \:=\:\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \left(\mathrm{3}{u}_{\mathrm{0}} −\mathrm{4}{u}_{\mathrm{1}} \right)…