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 axes are drawn to the ellipse x225+y216=1, then |α|<94

If the length of 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

a.the given equation is=x−1132+y−2132 =1a25x+12y−1132 It represents ellipse if 1a2<1⇒a2>1⇒a>1b. 4x2+8x+9y2−36y=−4 ⇒4x2+2x+1+9y2−4y+4=36⇒ (x+1)29+(y−2)24=1 Hence, (−1,2) is centre and (1,2) lies on the major axis Then required minimum distance is 1.c.Equation of normal at P(θ) is 5sec⁡θx−4cosecθy=9 and it passes through P(0, α) ∴ α =−94cosec⁡θ⇒ α =−94sin⁡θ⇒ |α| <94d. 2b2a=2a3⇒b2a2=13⇒1-e2=13⇒e=23

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