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Question

(c)

An ellipse passes through $(0,0)$, $(1,0)$ and $(2,0)$. Its minor and major axis are parallel to coordinates axes. One of its foci lies on $Y$-

Vedclass · Accepted Answer

$\sqrt{2}-1$

Vedclass · Answer

(c)

An ellipse passes through $(0,0)$, $(1,0)$ and $(2,0)$. Its minor and major axis are parallel to coordinates axes. One of its foci lies on $Y$-axis.

Let one foci is $(0, k)$.

$	herefore O B$ is the latusrectum of ellipse $F_1(0, k)$ is mid-point of $O B$

$F_1(0,1) \quad\left[\because k=\frac{2+0}{2}=1ight]$

Now $F_2(h, k)=F_2(h, 1)$

Now by definition of ellipse

$B F_1+B F_2=2 a =O F_1+O F_2$

$	herefore \quad B F_1+B F_2 =O F_1+O F_24$

$\sqrt{1+1}+\sqrt{(h-1)^2+1} =\sqrt{0+1}+\sqrt{h^2+1}$

$\sqrt{2}+\sqrt{(h-1)^2+1} =1+\sqrt{h^2+1}$

$\Rightarrow \quad h =1$

$	herefore F_1(0,1) 	ext { and } F_2(1,1)$

$F_1 F_2=2 a e =1$

$	herefore \quad O F_1+O F_2=2 a =1+\sqrt{2}$

$\Rightarrow \quad e=\frac{1}{\sqrt{2}+1} \Rightarrow e =\frac{\sqrt{2}-1}{2-1}$

$e =\sqrt{2}-1$

An ellipse with its minor and major axis parallel to the coordinate axes passes through $(0,0),(1,0)$ and $(0,2)$. One of its foci lies on the $Y$-axis. The eccentricity of the ellipse is

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