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Question-69092




Question Number 69092 by Mr. K last updated on 19/Sep/19
Answered by mr W last updated on 21/Sep/19
Commented by mr W last updated on 21/Sep/19
AX=(a/(cos θ))  XY=((a−a tan θ)/(cos θ))  AX−XY=((a tan θ)/(cos θ))=((a sin θ)/(cos^2  θ))  (√(AX^2 +(AX−XY)^2 ))=(√((a^2 /(cos^2  θ))+((a^2 sin^2  θ)/(cos^4  θ))))  =(√((a^2 (cos^2  θ+sin^2  θ))/(cos^4  θ)))=(a/(cos^2  θ))  ((AX^2 )/( (√(AX^2 +(AX−XY)^2 ))))=((((a/(cos θ)))^2 )/(a/(cos^2  θ)))=a  ⇒proved!
$${AX}=\frac{{a}}{\mathrm{cos}\:\theta} \\ $$$${XY}=\frac{{a}−{a}\:\mathrm{tan}\:\theta}{\mathrm{cos}\:\theta} \\ $$$${AX}−{XY}=\frac{{a}\:\mathrm{tan}\:\theta}{\mathrm{cos}\:\theta}=\frac{{a}\:\mathrm{sin}\:\theta}{\mathrm{cos}^{\mathrm{2}} \:\theta} \\ $$$$\sqrt{{AX}^{\mathrm{2}} +\left({AX}−{XY}\right)^{\mathrm{2}} }=\sqrt{\frac{{a}^{\mathrm{2}} }{\mathrm{cos}^{\mathrm{2}} \:\theta}+\frac{{a}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \:\theta}{\mathrm{cos}^{\mathrm{4}} \:\theta}} \\ $$$$=\sqrt{\frac{{a}^{\mathrm{2}} \left(\mathrm{cos}^{\mathrm{2}} \:\theta+\mathrm{sin}^{\mathrm{2}} \:\theta\right)}{\mathrm{cos}^{\mathrm{4}} \:\theta}}=\frac{{a}}{\mathrm{cos}^{\mathrm{2}} \:\theta} \\ $$$$\frac{{AX}^{\mathrm{2}} }{\:\sqrt{{AX}^{\mathrm{2}} +\left({AX}−{XY}\right)^{\mathrm{2}} }}=\frac{\left(\frac{{a}}{\mathrm{cos}\:\theta}\right)^{\mathrm{2}} }{\frac{{a}}{\mathrm{cos}^{\mathrm{2}} \:\theta}}={a} \\ $$$$\Rightarrow{proved}! \\ $$
Commented by TawaTawa last updated on 21/Sep/19
Wow, nice sir
$$\mathrm{Wow},\:\mathrm{nice}\:\mathrm{sir} \\ $$
Commented by Mr. K last updated on 21/Sep/19
Is there another way to prove  without using trigonometry?
$${Is}\:{there}\:{another}\:{way}\:{to}\:{prove} \\ $$$${without}\:{using}\:{trigonometry}? \\ $$
Commented by mr W last updated on 22/Sep/19
i think so. for example you can set  AB=a, BX=b.
$${i}\:{think}\:{so}.\:{for}\:{example}\:{you}\:{can}\:{set} \\ $$$${AB}={a},\:{BX}={b}. \\ $$

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