Question

Given distinct straight lines $OA$ and $OB.$ From a point in $OA$ a perpendicular is drawn to $OB;$ from the foot of this perpendicular a line is drawn perpendicular to $OA.$ From the foot of this second perpendicular a line is drawn perpendicular to $OB;$ and so on indefinitely. The lengths of the first and second perpendiculars are $a$ and $b,$ respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is:

$\dfrac b{a-b}$

$\dfrac a{a-b}$

$\dfrac{ab}{a-b}$

$\dfrac{b^2}{a-b}$

$\dfrac{a^2}{a-b}$

Answer: E

Concepts:geometric sequence similarity summation

Difficulty rating: 2190

Small Hint:

Successive right triangles formed by the two fixed lines are similar

Big Hint:

The perpendicular lengths form a geometric sequence with first term $a$ and ratio $\frac{b}{a}$

Solution:

Each new right triangle has the same acute angle, so the perpendicular lengths form a geometric sequence. Since the first two lengths are $a,b,$ the common ratio is $\frac{b}{a}.$ Convergence implies $0\lt \frac{b}{a}\lt1,$ and the sum is $\frac{a}{1-\frac{b}{a}}=\frac{a^2}{a-b}.$

Therefore, the correct answer is E.

1965 AMC 12 Problem 36

Answer: E

Difficulty rating: 2190

Small Hint:

Big Hint:

Solution:

Problem 36 in Other Years