Q. Reduce the equation x2 + y2 + 8x - 6y - 25 = 0 to the form AX2 + BY2 = K2 by shifting origin to a suitable point.

Dear Student,

  Let  the  origin  be  shifted  to  the  point  ( h ,  k ),  axes  remaining  parallel  to  the  original axes.  If  the  co-ordinates  ( x ,  y )  of  any  point  on  the  given  curve  change  to  (X,  Y),  thenx   =  X  +   h  and  y   =  Y  +   k . Substituting  for   x   and   y   in  the  given  equation,  the  transformed  equation  is(X+h)2+(Y+k)2+8(X+h)-6(Y+k)-25=0X2+h2+2Xh+Y2+k2+2Yk+8X+8h-6Y-6k-25=0X2+Y2+(2h+8)X+(2k-6)Y+h2+k2+8h-6k-25=0Since the given equation is to be reduced to the form AX 2  + BY 2  = K, the coefficients of X and Y  are  zerohence 2h+8=0 and 2k-6=0h=-4 and k=3hence  h2+k2+8h-6k-25   using value (h=-4 and k=3)=(-4)2+(3)2+8×-4-6×3-25=16+9-32-18-25=-50Hence, on shifting the origin to the point (-4, 3), the given equation reduces to X2  + Y2   50 = 0 i.e  X2+Y2= 50

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