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Question

Which of the following is/are true ? 


Correct Answer
A
There are infinite positive integral values of a for which (13x1)2+(13y2)2=(5x+12y1a)2 represents an ellipse
Correct Answer
B
The minimum distance of a point (1,2) from the ellipse 4x2+9y2+8x36y+4=0 is 1 unit
Correct Answer
C
If from a point P(0,α) two normals other than axes are drawn to the ellipse x225+y216=1, then |α|<94
Your Answer
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 (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
a.    The given equation is (x113)2+(y213)2=1a2(5x+12y113)2
It represents ellipse if 1a2<1a2>1a(,1)(1,)
Hence, there are infinite positive integrals possible.

b.   4x2+8x+9y236y=4 
4(x2+2x+1)+9(y24y+4)=36

(x+1)29+(y2)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 5secθx4cosecθ y=2516, 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=2a33b2=a2
From b2=a2(1e2),
 1=3(1e2)
e=23

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