Sliding ladder

Geometry Level 4

A ladder is placed vertically on a wall. If the ladder starts to slide against the wall to the floor, the general equation the midpoint of the ladder describes (considering that the wall is the $y$ -axis and the bottom of the ladder slides on the $x$ -axis) is given by

$Ax^2 + By^2 + Cxy + Dx + Ey = FL^n,$

where $A, B, C, D, E$ and $n$ are the lowest possible natural numbers, $F$ is some rational number and $L$ the ladder's length. Then

$A + B + C + D + E + F + n = N + \frac{1}{N},$

for some natural number $N.$ Find $N.$

The answer is 4.

1 solution

X X
Feb 24, 2019

Let the origin be $O$ , and the two ends of the ladder be $P,Q$ , the midpoint of the ladder be $M$

$\triangle POQ$ is always a right triangle, so the midpoint $M$ is the circumcenter of the triangle.

Hence $MP=MQ=MO=\frac{L}2$ . Since $MO$ is always $\frac L2$ , the equation is a circle.

The center is on the origin, and the radius is $\frac L2$

So the equation is $x^2+y^2=(\frac L2)^2=\frac{L^2}4$

$A+B+C+D+E+F+n=1+1+0+0+0+\frac14+2=4+\frac14, N=4$

Sliding ladder

The answer is 4.

1 solution

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