KVPY SX SB Exam 20141 Question Paper

Question : 5 of 120

Marks: +1, -0

Let C be the circle

x^ 2 + y^ 2 = 1

in the xy -plane. For each t > 0, let L

_t

be the line passing through (0,1) and (t,0) .
Note that L

_t

intersects C in two points, one of which is (0,1). Let Q

_t

be the other point. As t varies between 1 and

1 + √ 2

, the collection of points O, sweeps out an arc on C . The angle subtended by this arc at (0,0) is

Solution:

x^ 2 + y^2 = 1

L_t\;x/t+y/1=1

y=1-x/t

x^2+1+x^2/t^2-{2x}/t=1

x^2 (1+1/t^2)-{2x}/t=0

x = 0, y = 1
(0, 1)

x (1+1/t^2)={2 }/t

x={2t}/{t^2+1};y=1-2/{t^2+1}

y= {t^2-1}/{t^2+1}

Q_t({2t}/{ 1+t^2},{t^2-1}/{t^2+1})

1 ≤ t ≤ 1 +√ 2

\;\;t = \tan θ \ Q_t (\sin 2θ, – \cos 2θ)

θ ∈(45°,67 1°/2)\;\;\;\;

lies on circle C

so angle at centre

=π/ 4

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