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Question

# The tangent to the parabola y=x2+ax+1 at the point of intersection of y−axis also touches the circle x2+y2=r2 and no point of the parabola is below x−axis. Then

A
the radius of circle when a attains its maximum value is 110
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B
the slope of the tangent when radius of the circle is maximum is 0
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C
area bounded by the tangent and the coordinate axes is minimum when a=1
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D
the minimum area bounded by the tangent and the coordinate axes is 14 sq. unit
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Solution

## The correct options are B the slope of the tangent when radius of the circle is maximum is 0 D the minimum area bounded by the tangent and the coordinate axes is 14 sq. unity=x2+ax+1 Since, no point of the parabola is below x−axis, ∴D=a2−4≤0 Therefore, maximum possible value of a is 2. Equation of the parabola, when a=2, is y=x2+2x+1 It intersects y−axis at (0,1). ∴ Equation of the tangent at (0,1) is y=2x+1 Since, y=2x+1 touches the circle x2+y2=r2, ∴r=1√5 Equation of the tangent at (0,1) to the parabola y=x2+ax+1 is y−1=a(x−0) ⇒ax−y+1=0 As it touches the circle, ∴r=1√a2+1 Radius is maximum when a=0 Therefore, equation of the tangent, when a=0 is y=1. Hence, slope of the tangent is 0. Equation of tangent is y=ax+1. Intercepts are −1a and 1. Therefore, area of the triangle bounded by tangent and the axes is A=12∣∣∣−1a⋅1∣∣∣=12|a| Area is minimum, when a=2 And minimum area =14 sq. unit

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