The tangent at a point `P(acosvarphi,bsinvarphi)` of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` meets its auxiliary circle at two points, the chord joining which subtends a right angle at the center. Find the eccentricity of the ellipse.

Options

A. `(1)/(sqrt(1+cos^(2)theta))`
B. `(1)/(sqrt(1+sin^(2)theta))`
C. `sqrt(1+cos^(2)theta)`
D. `sqrt(1+sin^(2)theta)`

Answer

B

Explanation

The equation of the tangent at P`(theta)` to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` is
`(x)/(a)costheta+(y)/(b)sintheta=1`
The combined equation of the lines CQ and CR is
`x^(2)+y^(2)=a^(2)((x)/(a)costheta+(y)/(b)sintheta)^(2)`
The lines represented by this equation and at right angle.
`therefore 1-cos^(2)theta+1(a^(2))/(b^(2))sin^(2)theta=0`
`rArr sin^(2)theta+1-(1)/(1-e^(2))sin^(2)theta=0`
`rArr 1-e^(2)=(sin^(2)theta)/(1+sin^(2)theta)rArr e^(2)=(1)/(1+sin^(2)theta)rArr e=(1)/(sqrt(1+sin^(2)theta))`