The ellipse $x^{2}+4y^{2}= 4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is Option 1) $x^{2}+12y^{2}=16$ Option 2) $4x^{2}+48y^{2}=48$   Option 3) $4x^{2}+64y^{2}=48$ Option 4) $x^{2}+16y^{2}=16$

As we learnt in

Stanard equation -

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1$

- wherein

$a\rightarrow$ Semi major axis

$b\rightarrow$ Semi minor axis

Equation is  $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$

It passes through (2, 1)

Solving, $b^{2}=\frac{4}{3}$

Hence equation is

$\frac{x^{2}}{16}+\frac{3y^{2}}{4}=1$

$x^{2}+3y^{2}=16$

Option 1)

$x^{2}+12y^{2}=16$

Correct

Option 2)

$4x^{2}+48y^{2}=48$

Incorrect

Option 3)

$4x^{2}+64y^{2}=48$

Incorrect

Option 4)

$x^{2}+16y^{2}=16$

Incorrect

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