Tangent to the parabola `y=x^(2)+ax+1` at the point of intersection of the y-axis also touches the circle `x^(2)+y^(2)=r^(2)`. Also, no point of the parabola is below the x-axis.
The minimum area bounded by the tangent and the coordinate axes is

Options

A. 1
B. `1//3`
C. `1//2`
D. `1//4`

Answer

D

Explanation

(4) The equation of tangent is y=ax+1.
The intercepts are `-1//a` and 1.
Therefore, the area of the triangle bounded by tangent and the axes is
`(1)/(2)|-(1)/(a)*|=(1)/(2|a|)`
It is minimum when a=2. Therefore,
Minimum area `=(1)/(4)`