Show that the tangent at point P (cp, c/p) has the equation y p 2 + x = 2 c p. Theperpendicular line from the origin meets the tangent at point T.Find the locus T as p varies.

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Show that the tangent at point P (cp, c/p) has the equation   y      p          2        +  x  =  2  c  p. Theperpendicular line from the origin meets the tangent at point T.Find the locus T as p varies.

Cheyanne Charles · Accepted Answer

Show that the tangent atpoint P (cp, c/p) has the equation   y      p          2        +  x  =  2  c  p. The perpendicular line from the origin meets the tangent at point T. Find the locus T as pvaries.For point P x = cp and y = c/p or   x  y  =  (  c  p  )  (  c      /    p  )  =      c          2      .  x  y  =  (  c  p  )  (  c      /    p  )  =      c          2      [since x = cp]  ∴  (  d  y      /    d  x  )  =  −      c          2            /    (  c  p      )          2        =  −  1      /        p          2       which is the slope of thetangent.Hence, the equation of the tangent is   y  =  m  x  +  b  =  (  −  1      /        p          2        )  x  +  b .....(1)Since it passes through P, P must satisfy thisrelationship.Hence   (  c      /    p  )  =  (  −  1      /        p          2        )  (  c  p  )  +  b  =  −  (  c      /    p  )  +  b  ∴  b  =  2  c      /    pSubstituting in (1), we get   y  =  (  −  1      /        p          2        )  x  +  b  =  (  −  1      /        p          2        )  x  +  (  2  c      /    p  )Multiplying everything by       p          2      , we get  y      p          2        =  −  x  +  2  c  p or   y      p          2        +  x  =  2  c  p ......(2)A line passing through the origin has the equation y = mx since b = 0.Thus the slope of this line is m. But that slope must be thenegative reciprocal of the slope of the tangent.Hence equation of the perpendicular line from the origin mustbe   y  =      p          2        x. .....(3)Since this line meets the tangent at T, Co-ordinates of T mustsatisfy both the equations (2) and (3)   ∴  y      p          2        =  −  x  +  2  c  p and   y  =      p          2        x must hold.   ∴  y      p          2        =  (      p          2        x  )      p          2        =      p          4        x  =  −  x  +  2  c  p     o  r         p          4        x  +  x  =  2  c  p  =  (      p          4        +  1  )  x     o  r     x  =  2  c  p      /    (      p          4        +  1  ) ....(4) and   y  =      p          2        x  =  2  c      p          3            /    (      p          4        +  1  ) ....(5)Thus T is [2cp/ (p^{4}+1),2cp^{3}/ (p^{4}+1)].Dividing (5) by (4) we get   (  y      /    x  )  =      p          2

Show that the tangent at point P (cp, c/p) has the equation yp^2 + x = 2cp. The perpendicular line from the origin meets the tangent at point T.Find the locus T as p varies.

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