Question

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?

$0$

$2$

$4$

$8$

infinitely many

Solution:

The perimeter condition gives $CA+CB=50-10=40,$ so $C$ lies on an ellipse with foci $A,B$ and major axis $2a=40.$ Thus $a=20$ and $c=5,$ so the semi-minor axis is $b=\sqrt{a^2-c^2}=\sqrt{375}\approx19.36.$

For area $100$ with base $AB=10,$ the height from $C$ must be $\dfrac{2\cdot100}{10}=20.$ But the greatest possible height on the ellipse is $b\approx19.36\lt20,$ so no such $C$ exists.

Thus, A is the correct answer.

2019 AMC 12B Problem 6

Solution:

Problem 6 in Other Years