The tangent at point(θ) on the ellipse x2/a2+y2/b2=1 meets the auxiliary circle in two points which subtends

The tangent at point(θ) on the ellipse x2/a2+y2/b2=1 meets the auxiliary circle in two points which subtends a right angle at the centre,the eccentricity of the ellipse is

Tangent at P(θ) to the ellipse is : (x/a) cos θ + (y/b) sin θ = 1 ... (1)

If a > b, then the auxiliary circle is : x² + y = a² ........................ (2)

Joint equation of two lines joining centre of the

ellipse(origin) to the points of intersection of (1)

and (2) is obtained by making (2) homogeneous

with the help of (1) as follows :

... x² + y² = a² [ (x/a) cos θ + (y/b) sin θ ]²

Simplifying, this equation can be put in the form

...( 1 - cos² θ ) x² - 2(x/a)(y/b) sin θ cos θ + [ 1 - (a²/b²) sin² θ ] y² = 0.

If these two lines are at right angles, then

... ( coeff. of x² ) + ( coeff. of y² ) = 0

∴ ( 1 - cos² θ ) + [ 1 - (a²/b²) sin² θ ] = 0

∴ ( sin² θ ) + 1 - [ a² / (a²(1-e²) ] sin² θ = 0

∴ ( sin² θ ) { 1 - [ 1 / (1-e²) ] } = -1

∴ ( sin² θ )( 1 - e² - 1 ) = -( 1 - e² )

∴ e² ( 1 + sin² θ ) = 1

∴ e = 1 / √( 1 + sin² θ ) ........................................… Ans.
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