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Question Number 43921 by peter frank last updated on 17/Sep/18

prove that product of lengths of perpendiculars  from any point of hyperbola to its  asymptotes is constant

$${prove}\:{that}\:{product}\:{of}\:{lengths}\:{of}\:{perpendiculars} \\ $$$${from}\:{any}\:{point}\:{of}\:{hyperbola}\:{to}\:{its} \\ $$$${asymptotes}\:{is}\:{constant} \\ $$

Answered by math1967 last updated on 18/Sep/18

let equn. of hyperbola is(x^2 /a^2 )−(y^2 /b^2 )=1  equns.of asymptotes are bx−ay=0  and bx+ay=0 let any pt(asec∅,btanφ)  perpendicular from pt. to asymptotes  are p_1 ,p_2   ∴p_1 =((basecφ−abtanφ)/(√(b^2 −a^2 )))  p_2 =((basecφ+abtanφ)/(√(b^2 +a^2 )))  ∴p_1 ×p_2 =((a^2 b^2 (sec^2 φ−tan^2 φ))/(√(b^4 −a^4 )))                 =((a^2 b^2 )/(√(b^4 −a^4 )))=constant

$${let}\:{equn}.\:{of}\:{hyperbola}\:{is}\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$${equns}.{of}\:{asymptotes}\:{are}\:{bx}−{ay}=\mathrm{0} \\ $$$${and}\:{bx}+{ay}=\mathrm{0}\:{let}\:{any}\:{pt}\left({asec}\emptyset,{btan}\phi\right) \\ $$$${perpendicular}\:{from}\:{pt}.\:{to}\:{asymptotes} \\ $$$${are}\:{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} \:\:\therefore{p}_{\mathrm{1}} =\frac{{basec}\phi−{abtan}\phi}{\sqrt{{b}^{\mathrm{2}} −{a}^{\mathrm{2}} }} \\ $$$${p}_{\mathrm{2}} =\frac{{basec}\phi+{abtan}\phi}{\sqrt{{b}^{\mathrm{2}} +{a}^{\mathrm{2}} }} \\ $$$$\therefore{p}_{\mathrm{1}} ×{p}_{\mathrm{2}} =\frac{{a}^{\mathrm{2}} {b}^{\mathrm{2}} \left({sec}^{\mathrm{2}} \phi−{tan}^{\mathrm{2}} \phi\right)}{\sqrt{{b}^{\mathrm{4}} −{a}^{\mathrm{4}} }} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{{a}^{\mathrm{2}} {b}^{\mathrm{2}} }{\sqrt{{b}^{\mathrm{4}} −{a}^{\mathrm{4}} }}={constant}\: \\ $$

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