In an ellipse, x2/a2 + y2/b2 = 1, a>b, if the focal distances of a point on the ellipse ( the point is neither on X-axis nor on Y-axis ) are r1and r2, then find the length of the normal drawn at this point ( length of normal means the length of the portion of the normal between the point and the major axis ) ?

The equation of the ellipse is

x- axis is the major axis of the ellipse.

Let S(ae, 0) and S' (– ae, o) be the foci of the ellipse, where e is the eccentricity of the ellipse.

Let P(a cos θ, b sin θ) be point on the ellipse.

Given, r1 = a(1 – e cos θ) and r2 = a(1 + e cos θ)

Equation of normal at P(a cos θ, b sin θ) is ax sec θ – by cosec θ = a2 b2      ...(1)

On the x- axis, y = 0

ax sec θ – 0 = a2b2

∴ Normal intersect the major axis at

Length of normal

Thus, length of the normal is .

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