Faridpur, 12th bdmo , p9
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Four circles are aligned where they touch each other. P is a point on the circumference of the first circle; Q is he center of the fourth circle. PB is the tangent of the fourth circle which intersects the second circle in point A & B. The radius of all circle is 7 , and the length of AB can be expressed as a root b where a and b are both natural number. Find the value of a X b.
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Raiyan Jamil
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Re: Faridpur, 12th bdmo , p9
You have missed an information. That is: "PQ line passes through the centers of all four circles."
And by this, the value of a and b will be 2 and 40. So, a*b = 80.
And by this, the value of a and b will be 2 and 40. So, a*b = 80.
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Re: Faridpur, 12th bdmo , p9
Suppose,$T$ is the point in which $PB$ touches the fourth circle.$PQ=49,QT=7$.Suppose,$O$ is the center of the second circle.Draw a perpendicular from $O$ to $AB$.Suppose,it meets $AB$ at point $D$.
Now,$\Delta ODP\sim\Delta QTP$.$\therefore \frac{OD}{7}=\frac{21}{49}\Leftrightarrow OD=3.DB=\sqrt{49-9}=\sqrt{40}.\therefore AB=2\sqrt{40}=4\sqrt{10}$
$\therefore a \times b=80$ or $40$
Now,$\Delta ODP\sim\Delta QTP$.$\therefore \frac{OD}{7}=\frac{21}{49}\Leftrightarrow OD=3.DB=\sqrt{49-9}=\sqrt{40}.\therefore AB=2\sqrt{40}=4\sqrt{10}$
$\therefore a \times b=80$ or $40$
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