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Omni-MATH / Two given circles intersect in two points P and Q . Show how to construct a segment AB passing through P and terminating on the two circles such that AP⋅PB is a maximum. [asy] size…

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problem

Two given circles intersect in two points PP and QQ . Show how to construct a segment ABAB passing through PP and terminating on the two circles such that AP⋅PBAP\cdot PB is a maximum. [asy] size(150); defaultpen(fontsize(7)); pair A=(0,0), B=(10,0), P=(4,0), Q=(3.7,-2.5); draw(A--B); draw(circumcircle(A,P,Q)); draw(circumcircle(B,P,Q)); label("A",A,(-1,1));label("P",P,(0,1.5));label("B",B,(1,1));label("Q",Q,(-0.5,-1.5)); [/asy]
Plain-text mathematical notation (without MathML)
Two given circles intersect in two points P and Q . Show how to construct a segment AB passing through P and terminating on the two circles such that AP⋅PB is a maximum.
[asy] size(150); defaultpen(fontsize(7)); pair A=(0,0), B=(10,0), P=(4,0), Q=(3.7,-2.5); draw(A--B); draw(circumcircle(A,P,Q)); draw(circumcircle(B,P,Q)); label("A",A,(-1,1));label("P",P,(0,1.5));label("B",B,(1,1));label("Q",Q,(-0.5,-1.5)); [/asy]
Original LaTeX notation
Two given circles intersect in two points $P$ and $Q$ . Show how to construct a segment $AB$ passing through $P$ and terminating on the two circles such that $AP\cdot PB$ is a maximum.
[asy] size(150); defaultpen(fontsize(7)); pair A=(0,0), B=(10,0), P=(4,0), Q=(3.7,-2.5); draw(A--B); draw(circumcircle(A,P,Q)); draw(circumcircle(B,P,Q)); label("A",A,(-1,1));label("P",P,(0,1.5));label("B",B,(1,1));label("Q",Q,(-0.5,-1.5)); [/asy]

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Source and history

task_id
ab4e3440-61c0-5cfa-9c08-eb88db722c80
task_key
test--ab4e3440-61c0-5cfa-9c08-eb88db722c80
task_revision_id
1
config / split
/ test

Official source

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