A pair of tangents AC and BD of lengths a and b respectively are drawn at the extremities of the diameter AB of length d. BC and AD intersect each other at point P. If a, d and b are in G.P, then which of the following is/are correct?
A
P(≢A,≢B) lies on the circumference of the given circle
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B
P lies on the director circle of the given circle
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C
The maximum distance of P from AB is d
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D
The maximum area of ΔABP is d24 sq. units
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Solution
The correct options are AP(≢A,≢B) lies on the circumference of the given circle D The maximum area of ΔABP is d24 sq. units
tanθ=bd,tanϕ=ad tanθtanϕ=abd2 a,dandbare in G.P ⇒d=√ab⇒abd2=1 ∴tanθtanϕ=1 ⇒tanθ=cotϕ ⇒tanθ=tan(90∘−ϕ) ⇒θ+ϕ=90∘ ∴∠APB=180∘−(θ+ϕ)=90∘ Since, diameter subtends a right angle at point P, therefore P lies on the circumference of the circle.
Area of ΔABP is maximum when its altitude become the radius of the circle i.e., d2 ⇒Maximum area=12×d×d2=d24sq. units