`C` is the center of the circle with center (0, 1) and radius unity. `P` is the parabola `y=a x^2dot` The set of values of `a` for which they meet at a point other than the origin is `a >0` (b) `a in (0,1/2)` `(1/4,1/2)` (d) `(1/2,oo)`

Options

A. `agt0`
B. `ain(0,1//2)`
C. (1/4, 1/2)
D. `(1//2,oo)`

Answer

D

Explanation

(4) Putting `x^(2)=y//a` in the circle `x^(2)+(y-1)^(2)=1`, we get
`(y)/(a)+y^(2)-2y=0`
(Note that for `alt0` they cannot intersect other than the origin)
Hence, we get
`y=0ory=2-(1)/(a)`
Substituting `y=2-(1)/(a)` in `y=ax^(2)`, we get
`ax^(2)=2-(1)/(a)`
`orx^(2)=(2a-1)/(a^(2))lt0`
`or alt(1)/(2)`