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Column AColumn B(A) The angle between asymptotes of the hyperbola5x22xyy2+4x+6y+1=0 is tan1(mn) (where m,n are coprime), then m+n is                 (P) 2(B) Distance between the focii of the curve represented by the equation x=2+5cosθ and y=3+4sinθ is(Q) 5(C) The number of distinct normal possible from (114,14) to the parabola y2=4x is        (R) 6(D) The pair of straight lines represented byx2+y2+3xy+4x+y1=0 intersect at P. If Q and R are the points of intersection of the pair of lines with the x-axis and the area of the ΔPQR is A, then A24 is                                     (S) 7(T) 8

Which of the following is the only INCORRECT combination?
  1. (D)(Q) 
  2. (A)(S) 
  3. (B)(R)
  4. (C)(P) 


Solution

The correct option is B (A)(S) 
(A)
The combined equation of asymptotes of the hyperbola differs from the equation of asymptotes.
5x22xyy2+4x+6y+1=0
a=5, h=1, b=1
tanθ=2h2aba+b=264=32
m+n=5
(A)(Q)

(B)
For x=2+5cosθ and y=3+4sinθ,
(x2)225+(y3)216=0
e2=1b2a2=925
Focii distance =2ae=6
(B)(R)

(C)
For the parabola y2=4x,
The equation of the normal to the parabola is,
y+tx=2t+t3 at (t2,2t)
It passes through (114,14)
14=114t+2t+t3
(t1)(2t+1)2=0
Therefore, two distinct normals are possible.
(C)(P)

(D)
For x2+y2+3xy+4x+y1=0
differentiating with respect to x,
2x+3y+4=0(1)   
differentiating with respect to y,
3x+2y+1=0(2) 
     
Solving (1) and (2), we get
P(1,2)
These lines cut x-axis at Q and R
x2+4x1=0(x+2)2=5x=2±5
QR=25
Area
A=12×25×2=25
A24=5
(D)(Q)

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