Question:
PN is the ordinate of any point P on the hyperbola x^2/a^2-y^2/b^2=1. If Q divides AP in the ratio a^2:b^2, show that NQ is perpendicular to AP, where AA' the transverse axis of the hyperbola.
Answer:
Let the coordinate of point P be (x, y)
Let Q is the middle term.
From the figure,
Coordinate of Q is {(a + x)/2, y/2}
Again rom the figure, coordinate of Q is calculated as
{(a2 x + b2 a)/(a2 + b2 ), b2 y/(a2 + b2 )}
So, {(a + x)/2, y/2} = {(a2 x + b2 a)/(a2 + b2 ), b2 y/(a2 + b2 )}
Equate coefficient of y, we get
=> 1/2 = b2 /(a2 + b2 )
=> 2b2 = a2 + b2
=> a2 = b2
=> a = b
This shows that Q divides AP in equal parts.
So, QN is perpendicular to AP
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