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Tangent to the parabola y=x^(2)+ax+1 at ...

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 (a) 1 (b) 1/3 (c) 1/2 (d) 1/4

A

1

B

`1//3`

C

`1//2`

D

`1//4`

Text Solution

Verified by Experts

The correct Answer is:
D
(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)`
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