Question

# Column AColumn B(A) The angle between asymptotes of the hyperbola5x2−2xy−y2+4x+6y+1=0 is tan−1(√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+y−1=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?(D)→(Q) (A)→(S) (B)→(R)(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. 5x2−2xy−y2+4x+6y+1=0 a=5, h=−1, b=−1 tanθ=∣∣∣2√h2−aba+b∣∣∣=∣∣∣2√64∣∣∣=√32 ⇒m+n=5 (A)→(Q) (B) For x=2+5cosθ and y=3+4sinθ, (x−2)225+(y−3)216=0 e2=1−b2a2=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 ⇒(t−1)(2t+1)2=0 Therefore, two distinct normals are possible. (C)→(P) (D) For x2+y2+3xy+4x+y−1=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+4x−1=0⇒(x+2)2=5⇒x=−2±√5 QR=2√5 Area A=12×2√5×2=2√5 ⇒A24=5 (D)→(Q)

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