Find Distance between the points for which lines that pass through the point ` (1, 1)` and are tangent to the curve represent parametrically as `x = 2t-t^2` and `y = t + t^2`

Options

A. `(2sqrt(43))/(9)`
B. 2
C. 3
D. `(2sqrt(53))/(9)`

Answer

D

Explanation

`x=2t-t^(2)" (i)"`
and `y=t+t^(2)" (ii)"`
`(dy)/(dx)=(dy//dt)/(dx//dt)=(1+2t)/(2-2t)`
Slope of tangent using the point `(1,1)` and `(2t-t^(2),t+t^(2))" (iii)"`
Equating (ii) and (iii), `(t+t^(2)-1)/(2t-t^(2)-1)=(1+2t)/(2-2t)`
`rArr" "3t^(2)-4t+1=0`
`rArr" "t=1,(1)/(3)`
For `t=1`, point is `P(1,2)` and for `t=(1)/(3)`, point is Q`((5)/(9),(4)/(9))`
`therefore" "PQ=(2sqrt(53))/(9)`