Which of the following is/are true?

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There are infinite positive integral values of a for which (13x−1)2+(13y−2)2=5x+12y−1a2 represents an ellipse.

The minimum distance of a point (1,2) from the ellipse 4x2+9y2+8x−36y+4=0 is 1

If from a point P(0, α) two normals other than the axes are drawn to the ellipse x225+y216=1, then α < 9/4.

If the length of the latus rectum of an ellipse is one-third of its major axis, then its eccentricity is equal to 13

answer is A.

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Detailed Solution

(1) The given equation isx−1132+y−2132=1a25x+12y−1132It represents an ellipse if1a2<1 or a2>1 or a>1 (2) 4x2+8x+9y2−36y=−4 or 4x2+2x+1+9y2−4y+4=36 or (x+1)29+(y−2)24=1Hence, (-1, 2) is centre and (1, 2) lies on the major axis. Then the required minimum distance is 1.(3) The equation of normal at P(θ) is 5sec⁡θx−4cosec⁡θ y=25-16 , and it passes through p(0, α). Therefore,α=−94cosec⁡θ=−94sin⁡θ or |α|<94(4) 2b2a=2a3or 3b2 = a2 From b2=a21−e2, we have 1=31−e2 or e=2/3.

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