An ellipse whose major axis is parallel to x-axis is such that the segments of a focal chord are 1 and 3 units. The lines $ax + by + c = 0$ are the chords of the ellipse such that a, b, c are in A.P. and bisected by the point at which they are concurrent. The equation of auxiliary circle is $x^{2} + y^{2} + 2 αx + 2 βy - 2 α - 1 = 0$ . Then

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Length of latus rectum is 3

Eccentricity of the ellipse is $\frac{1}{2}$

The locus of perpendicular tangents to the ellipse is $x^{2} + y^{2} = 7$

Area of an auxillary circle is $2 π$

answer is B, D.

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Detailed Solution

a, b, c are in A.P $\Rightarrow ax + by + c = 0$ are concurrent at $(1, - 2)$
$∴$ center of auxiliary circle $= (- α, - β) = (1, - 2)$
Radius of auxiliary circle = 2
Length of major axis $= 4 = 2 a$
$∵ \frac{1}{SP} + \frac{1}{SQ} = \frac{2 a}{b^{2}} \Rightarrow b = \sqrt{3}$ , hence $e = \frac{1}{2}$