KVPY SA Exam 05-Nov-2017 Question Paper

Question : 61 of 80

Marks: +1, -0

Part - II

Mathematics

Let S be the circle in xy-plane which touches the x-axis at point A, the y-axis at point B and the unit circle

x _2 + y_2 = 1

at point C externally. If O denotes the origin, then the angle OCA equals

Solution:

∴ Slope of OM = 1
∴ ∠COA = 45°

C(1/√2,1/√2)

Now OM = OC + CM

√{2h} = 1+h

Squaring

2h^2 = h^2 + 1 + 2h

h^2 -2h -1 = 0

h={2±√8}/2⇒h=1±√2

A(1+√2,0)

Slope of AC =

{0-1/√2}/{1+√2-1/√2}

=-1/{√2+1}=-(√2-1)

-\tan22 {1/2}°=\tan(180-22 1/2)°

=\tan157 {1/2}°

∴ CAX=157 {1/2}°

∴ ∠OCA=157{1/2}°-45°

=112{1/2}°={5π}/8

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