First slide

Ellipse

Question

$The equation of the ellipse with its centre at (1,2), focus at (6,2) and passing through the point (4, 6) is$

Moderate

A

$\frac{(x - 1)^{2}}{45} + \frac{(y - 2)^{2}}{20} = 1$
B

$\frac{(x - 1)^{2}}{20} + \frac{(y - 2)^{2}}{45} = 1$
C

$\frac{(x - 1)^{2}}{25} + \frac{(y - 2)^{2}}{16} = 1$
D

$\frac{(x - 1)^{2}}{16} + \frac{(y - 2)^{2}}{25} = 1$

Solution

$Centre of the ellipse is C(1,2).$
$Also focus is at 5(6,2)$
$Since axis is parallel to x-axis, equation of ellipse is$
$\frac{(x - 1)^{2}}{a^{2}} + \frac{(y - 2)^{2}}{b^{2}} = 1$
$\begin{array}{l} It passes through (4, 6) \\ ∴ \frac{9}{a^{2}} + \frac{16}{b^{2}} = 1 \end{array}$
$\begin{array}{ll} Also a e = 5 \\ \Rightarrow b^{2} = a^{2} (1 - e^{2}) \\ \Rightarrow b^{2} = a^{2} - a^{2} e^{2} \\ \Rightarrow b^{2} = a^{2} - 25 \end{array}$
$\begin{array}{ll} ∴ \frac{9}{a^{2}} + \frac{16}{a^{2} - 25} = 1 \\ ∴ (a^{2} - 5) (a^{2} - 45) = 0 \\ ∴ a^{2} = 45 (as if a^{2} = 5, b^{2} = - 20) \\ ∴ b^{2} = 20 \end{array}$

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