KVPY 2019 SB SX Exam Previous Papers

Question : 4 of 120

Marks: +1, -0

Let AB be a line segment with midpoint C, and D be the midpoint of AC. Let C

_1

be the circle with diameter AB, and C

_2

be the circle with diameter AC. Let E be a point of C

_1

such that EC is perpendicular to AB. Let F be a point on C

_2

such that DF is perpendicular to AB, and E and F lie on opposite sides of AB. Then the value of sin ∠FEC is

Solution:

According to given informations in the question, if radius of circle

C_{1}

2 a

, centre

c

is at origin

(0,0)

and

A B

is along

X

-axis, then

B(2 a, 0), A(-2 a, 0), E(0,2 a)

and

F(-a,-a)

Let

∠ F E C=θ

Then, slope of line

E F=m=\tan (\;{π}/{2}-θ)

⇒ \cot θ=\;{2 a-(-a)}/{0-(-a)}=3 \;\;

[ ∵ m=\;{y_{2}-y_{1}}/{x_{2}-x_{1}}]

∴ \sin (∠ F E C)

\;\; =\sin θ=\;{1}/{√{1+\cot ^{2} θ}}v

=\;{1}/{√{1+9}}

=\;{1}/{√{10}}

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