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Category: Coordinate Geometry

A-normal-chord-to-an-ellipse-x-2-a-2-y-2-b-2-1-make-an-angle-of-45-with-the-axis-prove-that-the-square-of-its-length-is-equal-to-32a-4-b-4-a-2-b-2-3-

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Question Number 51921 by peter frank last updated on 01/Jan/19 $${A}\:{normal}\:{chord}\:{to}\:{an}\: \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$${make}\:{an}\:{angle}\:{of}\:\mathrm{45}^{°} \\ $$$${with}\:{the}\:{axis}.{prove} \\ $$$${that}\:{the}\:{square}\:{of}\:{its}\: \\ $$$${length}\:{is}\:{equal}\:{to}…

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Prove-that-the-two-parabola-y-2-4ax-and-y-2-4c-x-b-cannot-have-a-common-normal-other-than-the-axis-unless-b-a-c-gt-2-

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Question Number 51901 by peter frank last updated on 31/Dec/18 $${Prove}\:{that}\:{the}\:{two}\: \\ $$$${parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:{and} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{c}\left({x}−{b}\right)\:{cannot} \\ $$$${have}\:{a}\:{common}\:{normal} \\ $$$${other}\:{than}\:{the}\:{axis}\:{unless} \\ $$$$\frac{{b}}{{a}−{c}}>\mathrm{2} \\ $$…

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If-p-and-q-are-the-length-of-perpendicular-from-the-origin-to-the-lines-xcos-ysin-kcos2-and-xsec-ycosec-k-respectively-prove-that-p-2-4q-2-k-2-

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Question Number 51822 by peter frank last updated on 30/Dec/18 $${If}\:\:{p}\:{and}\:{q}\:\:{are}\:{the}\:{length} \\ $$$${of}\:{perpendicular}\:{from} \\ $$$${the}\:{origin}\:{to}\:{the}\:{lines} \\ $$$${x}\mathrm{cos}\:\theta−{y}\mathrm{sin}\:\:\theta={kcos}\mathrm{2}\theta \\ $$$${and}\:{x}\mathrm{sec}\:\theta+{y}\mathrm{cosec}\:\theta={k} \\ $$$${respectively} \\ $$$${prove}\:{that} \\ $$$${p}^{\mathrm{2}}…

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