The normal at a variable point `P` on the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` of eccentricity `e` meets the axes of the ellipse at `Qa n dRdot` Then the locus of the midpoint of `Q R` is a conic with eccentricity `e '` such that (a)`e^(prime)` is independent of `e` (b) `e^(prime)=1` (c)`e^(prime)=e` (d) `e^(prime)=1/e`

Options

A. e' is independent of e
B. e'=1
C. e'=e
D. `e'=1//e`

Explanation

Normal at point `P(a cos theta, b sin theta)` is
`(ax)/(cos theta)-("by")/(sin theta)=a^(2)-b^(2)" "`
It meets the axes at
`Q(((a^(2)-b^(2))cos theta)/(a),0)`
and `Q(0,((a^(2)-b^(2))sin theta)/(b))`
Let T(h,k) be a midpoint of QR. Then,
`2h=((a^(2)-b^(2))cos theta)/(b)`
and `2h=((a^(2)-b^(2))sin theta)/(b)`
or `cos ^(2)thetasin^(2)theta=(4h^(2)a^(2))/((a^(2)-b^(2))^(2))+(4k^(2)b^(2))/((a^(2)-b^(2))^(2))=1`
Therfore, the locus is
`(x^(2))/(((a^(2)-b^(2))^(2))/(4a^(2)))+(y^(2))/(((a^(2)-b^(2))^(2))/(4b^(2)))=1" "(2)`
Which is an ellipse, having eccentricity e' givne by
`e'^(2)=1-((a^(2)-b^(2))^(2)//4a^(2))/((a^(2)-b^(2))^(2)//4b^(2))=1-(b^(2))/(a^(2))=e^(2)`
`a'=e`
Note:
In (2), `((a^(2)-b^(2)))/(4a^(2))lt((a^(2)-b^(2)))/(4b^(2))`. Hence, the x-axis is the minor axis.