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 = a2 – b2
∴ Normal intersect the major axis at
Length of normal
Thus, length of the normal is .