The locus of the point which divides the double ordinates of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` in the ratio `1:2` internally is `(x^2)/(a^2)+(9y^2)/(b^2)=1` (b) `(x^2)/(a^2)+(9y^2)/(b^2)=1/9` `(9x^2)/(a^2)+(9y^2)/(b^2)=1` (d) none of these

Options

A. `(x^(2))/(a^(2))-(9y^(2))/(b^(2))=(1)/(9)`
B. `(x^(2))/(a^(2))+(9y^(2))/(b^(2))=1`
C. `(9x^(2))/(a^(2))+(9y^(2))/(b^(2))=1`
D. None of these

Answer

B

Explanation

Let P (acos`theta,bsintheta`)
`Q(acostheta-bsintheta)`
Let a point `R(h,k)` divides the line joining the points P and Q internally in the ratio 1:2.
Then, `PR:RQ=1:2`
Now, by division formula
`h=acosthetaimpliescostheta=(h)/(a)` . . . (i)
and `k=(b)/(3)sinthetaimpliessintheta=(3k)/(b)` . . (ii)
On squaring and addint eqs. (i)
and (ii), we get
`(h^(2))/(a^(2))=(9k^(2))/(b^(2))=1`
Hence, locus of R is `(x^(2))/(a^(2))+(9y^(2))/(b^(2))=1`.