First slide

Ellipse

Question

An ellipse has eccentricity 1/2 and a focus at the point P(1/2, 1). One of its directrix is the common tangent near to the point P, to the circle $x^{2} + y^{2} = 1$ and the hyperbola $x^{2} - y^{2} = 1$ the equation of the ellipse is

Moderate

A

$3 x^{2} + 4 y^{2} - 6 x - 8 y + 4 = 0$
B

$3 x^{2} + 4 y^{2} - 2 x - 8 y + 4 = 0$
C

$4 x^{2} + 3 y^{2} - 8 x - 6 y + 4 = 0$
D

$4 x^{2} + 3 y^{2} - 8 x - 2 y + 4 = 0$

Solution

Equation of a common tangent is x = ±1, nearer to $P (1 / 2, 1) is x = 1 .$
o the directrix of the ellipse is x = 1 and the focus is
$\begin{array}{l} (x - 1 / 2)^{2} + (y - 1)^{2} = 1 / 4 (1 - x)^{2} \\ \Rightarrow 3 x^{2} + 4 y^{2} - 2 x - 8 y + 4 = 0 \end{array}$

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