Analytical Geometry | From any point P on the hyperbola x 2 /a 2-askIITians

Jackie kumar kumar Grade: 11

        From any point P on the hyperbola x²/a² - y²/b² =1,three normals other than that at P are drawn.Find locus of the centroid of triangle formed by their feet.

8 years ago

Answers : (1)

Pratham Ashish

17 Points

										let p be ( a secØ , b tanØ )
let a point on hyperbola be (h,k)
so  ,  h^2 /a^2 ^{_{-  k^2 /b^2  = 1 ...........................(1)}}
^{_{eq of normal at this point  ,}}
^{_{a^2 x / h  + b^2 y /k =   a^2 + b^2 .........................(2)}}
^{_{since this normal passes trough P,}}
_{^{a^3 sec}}Ø / h  + b^3 tanØ /k =  a^2 +b^2  ..........................(3)
let   _{^{a^3 sec}}Ø = A ,    b^3 tanØ  = B   ,  a^2 +b^2 = C,......then
        A/h + B/k = C.....................................(4)
from eq 4 we get,
      h = Ak/(ck-B),,,.........put ineq.  (1)
we get
      a^2 * C^2 * k^4  -  2 * C*B * k^3 +  (........)   k^2 + (....)  k +  const. = 0
   from this eq .
              k1 +k2 +k3 +k4 =  - ( -  2 * C*B)/   a^2 * C^2                      {   k1 +k2 +k3 +k4 = - b/a}
                                        =  2B/C
                                        = 2  b^3 tanØ/ ( _{^{a^2 + b^2)}}
_{^{since  ( h4,k4) is point P,    k4 = b}}tanØ
                           k1 +k2 +k3  =   2  b^3 tanØ/ ( _{^{a^2 + b^2)    -  b}}tanØ
                                               = _{^{-  b}}tanØ * (a^2- b^2)/( _{^{a^2 + b^2)}}
 
 for centroid ,            y _c=  ( k1 +k2 +k3  )/3
                                      =    _{^{-  b}}tanØ * (a^2- b^2) / 3( _{^{a^2 + b^2)}}
_{^{in asimilar way we can get,}}
_{^x c =  ^{( h1 +h2+ h3) /3}}
_{^{= a}}secØ * (a^2- b^2) / 3( _{^{a^2 + b^2)}}
_^if(a^2- b^2) / 3( _{^{a^2 + b^2)  = G}}
_{^{(xc, yc )  =   aG}}secØ  ,   _{^{-  b  G}}tanØ 
 which emplies that the locus of the centrid will be also a hyperbola, 
                           x^2 / (aG)^2    -    y^2 / (bG)^2   =1