Question

# Which of the following is/are true ?

A
There are infinite positive integral values of a for which (13x1)2+(13y2)2=(5x+12y1a)2 represents an ellipse
B
The minimum distance of a point (1,2) from the ellipse 4x2+9y2+8x36y+4=0 is 1 unit
C
If from a point P(0,α) two normals other than axes are drawn to the ellipse x225+y216=1, then |α|<94
D
If the length of latus rectum of an ellipse is one-third of its major axis, then its eccentricity is equal to 13

Solution

## The correct options are A There are infinite positive integral values of a for which (13x−1)2+(13y−2)2=(5x+12y−1a)2 represents an ellipse B The minimum distance of a point (1,2) from the ellipse 4x2+9y2+8x−36y+4=0 is 1 unit C If from a point P(0,α) two normals other than axes are drawn to the ellipse x225+y216=1, then |α|<94a.    The given equation is (x−113)2+(y−213)2=1a2(5x+12y−113)2 It represents ellipse if 1a2<1⇒a2>1⇒a∈(−∞,−1)∪(1,∞) Hence, there are infinite positive integrals possible. b.   4x2+8x+9y2−36y=−4  ⇒4(x2+2x+1)+9(y2−4y+4)=36 ⇒(x+1)29+(y−2)24=1 Hence, (−1,2) is the centre and (1,2) lies on the major axis. Then required minimum distance is 1. c.    Equation of normal at P(θ) is 5secθx−4cosecθ y=25−16, and it passes through P(0,α) ∴α=−94cosec θ ⇒α=−94sinθ ⇒α∈(−94,94) because for sin90° or sin270° P(θ) will become end point of minor axis ⇒|α|<94 d.    Length of latus-rectum =13(length of major axis) 2b2a=2a3⇒3b2=a2 ⇒ From b2=a2(1−e2),  1=3(1−e2) ⇒e=√23

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