Let `d_1a n dd_2` be the length of the perpendiculars drawn from the foci `Sa n dS '` of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` to the tangent at any point `P` on the ellipse. Then, `S P : S^(prime)P=` `d_1: d_2` (b) `d_2: d_1` `d1 2:d2 2` (d) `sqrt(d_1):sqrt(d_2)`

Options

A. `d_(1):d_(2)`
B. `d_(2):d_(1)`
C. `d_(1)^(2)`
D. none of these

Answer

A

Explanation

Let `(x)/(a) cos theta +(y)/(b)sin theta=1` be a tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` at point P `(a cos theta, b sin theta)`. Then,
`d_(1)|(ecostheta-1)/(sqrt((cos^(2)theta)/(a^(2))+(sin^(2)theta)/(b^(2))))|and d_(2)|(ecostheta+1)/(sqrt((cos^(2)theta)/(a^(2))+(sin^(2)theta)/(b^(2))))|`
`rArr d_(1):d_(2)=|ecostheta-1| : |ecostheta+1|`
`rArr d_(1):d_(2)=|aecostheta-a| : |aecostheta+a|`
`rArr d_(1):d_(2)=SP:S'P`.