Loading [MathJax]/jax/output/HTML-CSS/jax.js


1)

Consider the hyperbola H: x2-y2=1 and a circle S with centre N(x2,0). Suppose that H and S touch each other at a point P(x1,y1) with x1>1 and y1>0. The common tangent to H and  S at P intersects the X-axis  at point M.  If  (l,m) is the centroid of PMN, then the correct expression(s) is/are


A) dldx1=113x21 for x1>0

B) dmdx1=x13(x211 for x1>0

C) dldx1=1+13x21 for x1>0

D) dmdy1=13 for y1>0

Answer:

Option A,B,D

Explanation:

 1732021908_p12.PNG

Equation of family of circles touching hyperbola at (x1,y1) is

 (x-x1)2+(y-y1)2+λ(x x1-y y1-1)=0

 Now, its centre is (x2,0).

  [(λx12x1)2,(2y1λy1)2]=(x2,0)

        2y1+λy1=0λ=2

 and 2x1λx1=2x2x2=2x1

        P(x1,x211)

 And    N(x2,0)=(2x1,0)

  As tangent intersect X-axis at  M(1x,0)

  Centroid of   PMN=(l,m)

       (3x11x13,y1+0+03)=(l,m)

  l=3x1+1x13

 On differentiating w.r.t x1, we get

     dldx1=31x213

         dldx1=113x21, for   x1>1

 and             m=x2113

    On differentiating w.r.t x1  , we get

dmdx1=2x12×3x211=x13x211,        for x1>1   

also    m=y13

  On differentiating  w.r.t y1, we get

      dmdy1=13,    for   y1>0