A curve is given by the equations `x=sec^2theta,y=cotthetadot` If the tangent at `Pw h e r etheta=pi/4` meets the curve again at `Q ,t h e n[P Q]` is, where [.] represents the greatest integer function, _________.

Answer

3

Explanation

`(dy)/(dx)=(-1)/2cot^(3)theta-(-1)/2` at `theta=(pi)/4`
Also, the point `p` for `theta=(pi)/4` is `(2,1)` equation of tangen is`y-1=(-1)/2(x-2)`
This meets the curve whose ccartesian equation on eliminating `theta` by `sec^(2)theta-tan^(2)theta=1` is
`y^(2)=1/(x-1)`
solving (1) and (2), we get `y=1, (-1)/2`
`:.x=2,5`
Hence `P` is `(2,1)` given and `Q` is `(5,(-1)/2)`
Therefore `PQ=-sqrt(45/4)=(3sqrt(5))/2 [PQ]=3`