Track / Stage 4 / 33 of 340 #293 of 1964

Problem 293

AMC 12 late, AIME early

Geometry Difficulty 4.5 Find the answer

6. A line $l$ is drawn through point $P(2,1)$ , intersecting the two coordinate axes at points $A$ and $B$ . When the area $S$ of $\triangle A O B$ takes values in $(0,+\infty)$ , the number of lines $l$ that can be drawn is ( ) lines.

Pick one

Official solution

6.D.

The line $l$ passes through $P(2,1)$ and forms a triangle with the two coordinate axes in the first quadrant, the minimum area of which is 4. In fact, let the equation of $l$ be
$\frac{x}{a}+\frac{y}{b}=1(a>0, b>0),$

then $\frac{2}{a}+\frac{1}{b}=1 \geqslant 2 \sqrt{\frac{2}{a} \cdot \frac{1}{b}}, a b \geqslant 8$ .
Thus, $S=\frac{1}{2} a b \geqslant 4$ .
When $S=4$ , there are 4 lines.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.