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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Circle

Olympiad workbook · IOQM · Circle

A light source at the point (0,16) in the coordinate plane casts...

A light source at the point

(0,16)

in the coordinate plane casts light in all directions. A disc (a circle along with its interior) of radius 2 with center at

(6,10)

casts a shadow on the

X

axis. The length of the shadow can be written in the form

m \sqrt{n}

where

m, n

are positive integers and

n

is square-free. Find

m+n

.

A light source at the point (0,16) in the coordinate plane casts light in all directions. A disc (a circle along with

Correct answer

21

Step-by-step solution

A light source at the point (0,16) in the coordinate plane casts light in all directions. A disc (a circle along with

C \equiv(x-6)^2+(y-10)^2=2^2

Let tangent be

m x-y+16=0

\begin{aligned} & \left|\frac{6 m+6}{\sqrt{1+m^2}}\right|=2 \\ & \Rightarrow 9 m^2+9+18 m=m^2+1 \\ & \Rightarrow 8 m^2+18 m+8=0 \\ & \Rightarrow 4 m^2+9 m+4=0 \\ & \Rightarrow m=\frac{-9 \pm \sqrt{17}}{8} \\ & \quad\left|x_1-x_2\right|=\left|\frac{16 \cdot 8}{-9+\sqrt{17}}-\frac{16 \cdot 8}{-9-\sqrt{17}}\right| \\ & \quad=\frac{128|-9-\sqrt{17}+9-\sqrt{17}|}{(81-17)} \\ & \quad=2|2 \sqrt{17}|=4 \sqrt{17} \\ & \Rightarrow m+n=21 \end{aligned}

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