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Question

# There are two points A(a,0) and B(b,0) on positive x-axis, where 0<a<b. Let P(0,y) be a variable point on positive y-axis. Then the maximum value of ∠APB occurs when y is

A
Arithmetic mean of a and b
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B
Geometric mean of a and b
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C
Harmonic mean of a and b
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D
y3=ab
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Solution

## The correct option is B Geometric mean of a and b From fig. we need to find the maximum value of θ tan(α+θ)=tanα+tanθ1−tanα tanθ ⇒by=ay+tanθ1−aytanθ ⇒by−aby2tanθ=ay+tanθ ⇒(1+aby2)tanθ=b−ay ⇒tanθ=(b−a)yy2+ab Let f=tanθ=(b−a)yy2+ab For θ to be maximum, f should also be maximum. dfdy=(y2+ab)(b−a)−(b−a)y(2y)(y2+ab)2=0 ⇒y2+ab−2y2=0⇒y2=ab ...(i) dfdy=(ab−y2)(b−a)(y2+ab)2 d2fdy2∣∣∣y2=ab =(y2+ab)2(b−a)(−2y)−(b−a)(ab−y2)2(y2+ab)2y(y2+ab)4 =(2ab)2(b−a)(−2√ab)(2ab)4 =−(b−a)(2√ab)(2ab)2 ∴d2fdy2<0 Hence, at y2=ab, θ→ maximum ∴y is G.M. of a,b.

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