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Question-28565




Question Number 28565 by ajfour last updated on 27/Jan/18
Answered by ajfour last updated on 27/Jan/18
eq. of tangents through P(h,k)     y−k = m(x−h)  applying condition for tangency:  (k−mh)^2 =a^2 m^2 +b^2   ⇒(a^2 −h^2 )m^2 −(2hk)m+b^2 −k^2 =0  m_1 +m_2 =((2hk)/(a^2 −h^2 ))  , m_1 m_2 =((b^2 −k^2 )/(a^2 −h^2 ))  ∣tan θ∣=∣((m_1 −m_2 )/(1+m_1 m_2 ))∣            =((√((m_1 +m_2 )^2 −4m_1 m_2 ))/(∣1+m_1 m_2 ∣))      =((√(4h^2 k^2 −4(b^2 −k^2 )(a^2 −h^2 )))/(∣a^2 −h^2 +b^2 −k^2 ∣))  ∣tan θ∣ =tan^(−1) ∣((2(√(b^2 h^2 +a^2 k^2 −a^2 b^2 )))/(a^2 +b^2 −(h^2 +k^2 )))∣    θ = tan^(−1) ((2ab(√((h^2 /a^2 )+(k^2 /b^2 )−1)))/((h^2 +k^2 )−(a^2 +b^2 ))) .
$${eq}.\:{of}\:{tangents}\:{through}\:{P}\left({h},{k}\right) \\ $$$$\:\:\:{y}−{k}\:=\:{m}\left({x}−{h}\right) \\ $$$${applying}\:{condition}\:{for}\:{tangency}: \\ $$$$\left({k}−{mh}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} {m}^{\mathrm{2}} +{b}^{\mathrm{2}} \\ $$$$\Rightarrow\left({a}^{\mathrm{2}} −{h}^{\mathrm{2}} \right){m}^{\mathrm{2}} −\left(\mathrm{2}{hk}\right){m}+{b}^{\mathrm{2}} −{k}^{\mathrm{2}} =\mathrm{0} \\ $$$${m}_{\mathrm{1}} +{m}_{\mathrm{2}} =\frac{\mathrm{2}{hk}}{{a}^{\mathrm{2}} −{h}^{\mathrm{2}} }\:\:,\:{m}_{\mathrm{1}} {m}_{\mathrm{2}} =\frac{{b}^{\mathrm{2}} −{k}^{\mathrm{2}} }{{a}^{\mathrm{2}} −{h}^{\mathrm{2}} } \\ $$$$\mid\mathrm{tan}\:\theta\mid=\mid\frac{{m}_{\mathrm{1}} −{m}_{\mathrm{2}} }{\mathrm{1}+{m}_{\mathrm{1}} {m}_{\mathrm{2}} }\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:=\frac{\sqrt{\left({m}_{\mathrm{1}} +{m}_{\mathrm{2}} \right)^{\mathrm{2}} −\mathrm{4}{m}_{\mathrm{1}} {m}_{\mathrm{2}} }}{\mid\mathrm{1}+{m}_{\mathrm{1}} {m}_{\mathrm{2}} \mid} \\ $$$$\:\:\:\:=\frac{\sqrt{\mathrm{4}{h}^{\mathrm{2}} {k}^{\mathrm{2}} −\mathrm{4}\left({b}^{\mathrm{2}} −{k}^{\mathrm{2}} \right)\left({a}^{\mathrm{2}} −{h}^{\mathrm{2}} \right)}}{\mid{a}^{\mathrm{2}} −{h}^{\mathrm{2}} +{b}^{\mathrm{2}} −{k}^{\mathrm{2}} \mid} \\ $$$$\mid\mathrm{tan}\:\theta\mid\:=\mathrm{tan}^{−\mathrm{1}} \mid\frac{\mathrm{2}\sqrt{{b}^{\mathrm{2}} {h}^{\mathrm{2}} +{a}^{\mathrm{2}} {k}^{\mathrm{2}} −{a}^{\mathrm{2}} {b}^{\mathrm{2}} }}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −\left({h}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)}\mid \\ $$$$\:\:\theta\:=\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{2}{ab}\sqrt{\frac{{h}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{k}^{\mathrm{2}} }{{b}^{\mathrm{2}} }−\mathrm{1}}}{\left({h}^{\mathrm{2}} +{k}^{\mathrm{2}} \right)−\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)}\:. \\ $$

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