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Question-49526




Question Number 49526 by ajfour last updated on 07/Dec/18
Commented by ajfour last updated on 07/Dec/18
If both the coloured areas are equal,  find equation of parabola in terms  of ellipse parameters a and b.
$${If}\:{both}\:{the}\:{coloured}\:{areas}\:{are}\:{equal}, \\ $$$${find}\:{equation}\:{of}\:{parabola}\:{in}\:{terms} \\ $$$${of}\:{ellipse}\:{parameters}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}. \\ $$
Answered by ajfour last updated on 07/Dec/18
y= Ax^2 −b    ,     (x^2 /a^2 )+(y^2 /b^2 ) = 1  (x^2 /a^2 )+(((Ax^2 −b)^2 )/b^2 ) = 1  A^2 x^4 +((b^2 /a^2 )−2Ab)x^2  = 0  ⇒  x_P  = ((√(2Ab−(b^2 /a^2 )))/A)  ⇒ ∫_0 ^(  x_P )  (b(√(1−(x^2 /a^2 )))−Ax^2 +b)dx =((πab)/4) .
$${y}=\:{Ax}^{\mathrm{2}} −{b}\:\:\:\:,\:\:\:\:\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{\left({Ax}^{\mathrm{2}} −{b}\right)^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:=\:\mathrm{1} \\ $$$${A}^{\mathrm{2}} {x}^{\mathrm{4}} +\left(\frac{{b}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\mathrm{2}{Ab}\right){x}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\Rightarrow\:\:{x}_{{P}} \:=\:\frac{\sqrt{\mathrm{2}{Ab}−\frac{{b}^{\mathrm{2}} }{{a}^{\mathrm{2}} }}}{{A}} \\ $$$$\Rightarrow\:\int_{\mathrm{0}} ^{\:\:{x}_{{P}} } \:\left({b}\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }}−{Ax}^{\mathrm{2}} +{b}\right){dx}\:=\frac{\pi{ab}}{\mathrm{4}}\:. \\ $$$$ \\ $$

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