In an ellipse, x^{2}/a^{2} + y^{2}/b^{2} = 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 r_{1}and r_{2}, 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

âˆ´ *x*- axis is the major axis of the ellipse.

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

Let P(*a* cos Î¸, *b* sin Î¸) be point on the ellipse.

Given, *r*_{1} = *a*(1 â€“ *e* cos Î¸) and *r*_{2} = *a*(1 + *e* cos Î¸)

Equation of normal at P(*a* cos Î¸, *b* sin Î¸) is *ax* sec Î¸ â€“ *by* cosec Î¸ = *a*^{2 }â€“ *b*^{2 }...(1)

On the *x- *axis, *y* = 0

âˆ´ *ax* sec Î¸ â€“ 0 = *a*^{2} â€“ *b*^{2}

âˆ´ Normal intersect the major axis at

Length of normal

Thus, length of the normal is .

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