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જે ઉપવલયની નાભિઓ $(-1, 0)$ અને $(7, 0)$ અને ઉત્કેન્દ્રતા $1/2$ હોય, તે ઉપવલય પરના બિંદુનું પ્રચલ સ્વરૂપ :
$\left( {3\, + \,\,8\cos \theta ,\,\,4\sqrt 3 \cos \,\,\theta } \right)$
$\left( {3\, + \,\,8\cos \theta ,\,\,4\sqrt 3 \sin \,\,\theta } \right)$
$\left( {3\, + \,\,4\sqrt 3 \cos \theta ,\,\,8\sin \,\,\theta } \right)$
એકપણ નહિ
Solution
Using Midpoint formula $X =\left(\frac{ x _1+ x _2}{2}\right)$ and $Y =\left(\frac{ y _1+ y _2}{2}\right)$
Center of the ellipse is the mid point of foci i.e $\left(\frac{-1+7}{2}+\frac{0+0}{2}\right)$ which is $(3,0)$
Distance Formula $=\sqrt{\left( x _2- x _1\right)^2+\left( y _2- y _1\right)^2}$
Now, Distance between the foci $=\sqrt{(7+1)^2+0}=8=2 ae \Rightarrow ae =4 \Rightarrow a =8$ since $e=\frac{1}{2}$ (given)
So $b =8 \sqrt{1- e ^2}=8 \sqrt{1-1 / 4}=4 \sqrt{3}$
Hence equation of ellipse is given by,
$\Rightarrow \frac{( x -3)^2}{8^2}+\frac{( y -0)^2}{(4 \sqrt{3})^2}=1$
So parametric points is, $x-3=8 \cos \theta$ and $y-0=4 \sqrt{3} \sin \theta$
$\Rightarrow x =3+8 \cos \theta, y =4 \sqrt{3} \sin \theta$
