Coordinate Geometry: all problems

The interior of a quadrilateral is bounded by the graphs of $(x + ay)^2 = 4a^2$ and $(ax - y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?

A laser is placed at the point $(3, 5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7, 5)$. What is the total distance the beam will travel along this path?

The point $P(a, b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1, 5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6, 3)$. What is $b - a$?

The line segment from $A(1, 2)$ to $B(3, 3)$ can be transformed to the line segment from $A'(3, 1)$ to $B'(4, 3)$, sending $A$ to $A'$ and $B$ to $B'$, by a rotation centered at the point $P(s, t)$. What is $|s - t|$?

What is the length of the boundary of the region in the $xy$-plane consisting of points of the form $(2u - 3w, v + 4w)$ with $0 \le u \le 1$, $0 \le v \le 1$, and $0 \le w \le 1$?

The area of the region bounded by the graph of

is $m + n\pi$, where $m$ and $n$ are integers. What is $m + n$?

A circle of radius $r$ is surrounded by three circles, whose radii are $1$, $2$, and $3$, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below. What is $r$?

Let $ABCD$ be a rhombus whose vertices all lie on the hyperbola $\frac{x^2}{20} - \frac{y^2}{24} = 1$ and are in that order. If its diagonals intersect at the origin, find the largest number less than $BD^2$ for all rhombuses $ABCD$.

The line $12x + 5y = 60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

Triangle $ABC$ has vertices $A = (3, 0)$, $B = (0, 3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$?

The lines with equations $ax - 2y = c$ and $2x + by = -c$ are perpendicular and intersect at $(1, -5)$. What is $c$?

The point $P = (1, 2, 3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated by $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated by $5$ units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$?

Line segment $\overline{AB}$ is a diameter of a circle with $AB = 24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

Right triangle $ABC$ has side lengths $BC = 6$, $AC = 8$, and $AB = 10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?

Two lines with slopes $\frac{1}{2}$ and $2$ intersect at $(2, 2)$. What is the area of the triangle enclosed by these two lines and the line $x + y = 10$?

A triangle has vertices $(0, 0)$, $(1, 1)$, and $(6m, 0)$, and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$?

Let $A$, $B$ and $C$ be three distinct points on the graph of $y = x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?

In rectangle $ABCD$, we have $A = (6, -22)$, $B = (2006, 178)$, $D = (8, y)$, for some integer $y$. What is the area of rectangle $ABCD$?

Find the area of the region enclosed by the graph of $|x - 60| + |y| = \left|\frac{x}{4}\right|$.

Find the number of ordered pairs $(x, y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60$.

Consider the parallelogram with vertices $(10, 45)$, $(10, 114)$, $(28, 153)$, and $(28, 84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A = (24, -1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B = (5, 6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14, 27)$ in the original system has coordinates $(\alpha, \beta)$ in the new coordinate system. Find $\alpha + \beta$.

Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12, 19)$ and $(23, 20)$, respectively, and the coordinates of $A$ are $(p, q)$. The line containing the median to side $\overline{BC}$ has slope $-5$. Find the largest possible value of $p + q$.

In rectangle $ABCD$, we have $AB = 8$, $BC = 9$, $H$ is on $\overline{BC}$ with $BH = 6$, $E$ is on $AD$ with $DE = 4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $\overline{GF} \perp \overline{AF}$. Find the length $GF$.