First slide

Ellipse

Question

$The slopes of the common tangents of the ellipse \frac{x^{2}}{4} + \frac{y^{2}}{1} = 1 and the circle x^{2} + y^{2} = 3 are$

Moderate

A

$\pm 1$
B

$\pm \sqrt{2}$
C

$\pm \sqrt{3}$
D

$None of these$

Solution

$Let m be the slope of the common tangent. Then,$
$\pm \sqrt{3} \sqrt{1 + m^{2}} = \pm \sqrt{4 m^{2} + 1}$
$or 3 + 3 m^{2} = 4 m^{2} + 1$
$\begin{array}{l} or m^{2} = 2 \\ or m = \pm \sqrt{2} \end{array}$

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