Let P be a point in the plane and C a circle with centre O and radius r. Then we define the power of P with respect to C as power of P with respect to C = O P 2 − r 2 (a) Determine the sign of the power of P with respect to C when (i) P lies inside C ; (ii) P lies on C ; (iii) P lies outside C . In parts (b) and (c) we regard distances as directed distances; that is, distances along a line in one direction have a positive sign associated with their length and distances in the opposite direction have a negative sign associated with their length. (b) If P lies inside C and a line through P meets C at two distinct points A and B. prove that power of P with respect to C = P A P B . (c) If P lies outside C, a line through P meets C at two distinct points A and B, and PT is one of the tangents from P to C. prove that power of P with respect to C = PA PB = PT. (d) If C has equation x 2 + y 2 + f x + g y + h = 0 and P has coordinates (x, y). find the power of P with respect to C in terms of x, y,f. g and h.

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Let P be a point in the plane and C a circle with centre O and radius r.     Then we define the power of P with respect to C as     power of P with respect to   C  =  O      P    2    −      r    2       (a) Determine the sign of the power of P with respect to C when     (i) P lies inside   C  ;     (ii) P lies on   C  ;     (iii) P lies outside   C  .     In parts (b) and (c) we regard distances as directed distances; that is, distances along a line in one direction have a positive sign associated with their length and distances in the opposite direction have a negative sign associated with their length.     (b) If P lies inside C and a line through P meets C at two distinct points A and B. prove that     power of P with respect to   C  =  P  A  P  B  .     (c) If P lies outside C, a line through P meets C at two distinct points A and B, and PT is one of the tangents from P to C. prove that     power of P with respect to C = PA PB = PT.     (d) If C has equation      x    2    +      y    2    +  f  x  +  g  y  +  h  =  0   and P has coordinates (x, y). find the power of P with respect to C in terms of x, y,f. g and h.

cindysnifflesuz · Accepted Answer

a) i) P lies inside COP &amp;lt; rso power is negativeii) P lies on COP= rPower is zeroiii) op&amp;gt;rPower is positive  d  )      x    2    +  2  f  x  +      f    2    +      y    2    +  2  g  y  +      g    2    =  f  x  +  g  y  +      f    2    +      g    2    −  h  (  x  +  f      )    2    +  (  y  +  g      )    2    =  f  x  +  g  y  +      f    2    +      g    2    −  h    c  e  n  t  e  r  =  (  −  f  ,  −  g  )    O      P    2    =  (  x  +  f      )    2    +  (  y  +  g      )    2          r    2    =  f  x  +  g  y  +      f    2    +      g    2    −  h    P  o  w  e  r  =  O      P    2    −      r    2      =  (  x  +  f      )    2    +  (  y  +  g      )    2    −  f  x  −  g  y  −      f    2    −      g    2    +  h    =      x    2    +      y    2    +  f  x  +  g  y  +  h  .  .  .  .  .  .  .  .  .  .  .  .  a  n  s

Let P be a point in the plane and C a circle with centre O and radius r^2 Determine the sign of the power of P with respect to C when...

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