First slide

Ellipse

Question

$The equation of ellipse circumscribing the quadrilateral whose sides are given by x =±2 and y = ±4 and$
$distance between foci is 4 \sqrt{6}, is$

Moderate

A

$\frac{x^{2}}{32} + \frac{y^{2}}{8} = 1$
B

$\frac{x^{2}}{8} + \frac{y^{2}}{32} = 1$
C

$\frac{x^{2}}{16} + \frac{y^{2}}{8} - 1$
D

$\frac{x^{2}}{8} + \frac{y^{2}}{16} = 1$

Solution

$Let equation be \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 where a^{2} = b^{2} (1 - e^{2})$
$Ellipse passes through the point (2,4).$
$\begin{array}{l} ∴ \frac{4}{a^{2}} + \frac{16}{b^{2}} = 1 - - - (i) \\ Also distance between foci is 4 \sqrt{6} . \end{array}$
$\begin{array}{l} ∴ b e = 2 \sqrt{6} \\ ∴ \frac{4}{b^{2} - 24} + \frac{16}{b^{2}} = 1 \\ \Rightarrow b^{4} - 44 b^{2} + 16 \times 24 = 0 \\ \Rightarrow (b^{2} - 32) (b^{2} - 12) = 0 \\ \Rightarrow b^{2} = 32 (as b > 4) \\ \Rightarrow a^{2} = 8 \end{array}$

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