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a double ordinate of the parabola y^2=2ax is of lenght 4a prove that the line joining the vertex to ends are at right angle

a double ordinate of the parabola y^2=2ax is of lenght 4a prove that the line joining the vertex to ends are at right angle

Grade:12th pass

1 Answers

Vikas TU
14149 Points
3 years ago
Double ordinate is perpendicular to axis of parabola hence the ordinates of double ordinate be
P(at2 ,2at ) and P1=(at2 ,-2at)
PP1 = 4a
  • ( (at2 – at2 )^2 + ( 2at+2at) ^2 ) ^1/2 = 4a
  • ( (4at) ^ 2) ^1/2 = 4a
  • 4at = 4a
  • t = 1;
Hence P = (at2 ,2at ) = ( a,2a)
P1 = ( a, -2a)
Vertex is (0,0)
Slope of OP = m1 = (a-0)/(2a-0) = ½.
Slope of OP1 = m2 = (a-0)/(-2a-0) = -½.
m1 .  m2 = ½ . -1/2
m1 . m2 = -1.
Angle POP1 = 900
Hence  line joining the vertex to ends are at right angles.

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