Coordinate Geometry

A line segment has a midpoint at $(4, 4)$ and one endpoint at $(0, 0)$. What is the gradient of this line segment?

A line has a gradient of $\frac{1}{2}$. If it passes through $(4, 6)$, find its equation in the form $y = mx + c$.

A line passes through $(0, 0)$ and has a gradient of $m$. Another line passes through $(0, 0)$ and is perpendicular to it. What is the gradient of the second line?

Find the coordinates of the midpoint of the line joining the points $(x, y)$ and $(3x, 5y)$.

A line segment $PQ$ has its midpoint at $M(3, 1)$. The point $P$ has coordinates $(k, 5)$ and the gradient of the line segment $PQ$ is $-2$. Find the value of the constant $k$.

The coordinates of the vertices of a parallelogram are $A(1, 2)$, $B(4, 2)$, $C(5, 5)$, and $D(2, 5)$. What is the midpoint of the diagonal $BD$?

If the gradient of the line joining $(x, 2)$ and $(5, 8)$ is $2$, find $x$.

Three vertices of a parallelogram are $A(1, 2)$, $B(4, 2)$, and $C(5, 5)$. Find the coordinates of the fourth vertex $D$ such that $ABCD$ is the parallelogram, and then determine the gradient of the diagonal $BD$.

A line segment has midpoint $(2, 1)$ and one endpoint at $(x, x^2)$. If the gradient of the segment is $1$, find the possible value of $x$ (where $x > 0$).

The coordinates of two points are $G(3, 4)$ and $H(x, y)$. If the midpoint of $GH$ is the origin $(0, 0)$, what is the gradient of the line $GH$?

A horizontal line passes through the point $(3, -5)$. What is the equation of this line?

Identify the $y$-intercept of the line represented by the equation $2x + 5y = 10$.

The diagram shows a line segment $AB$ where $A$ is $(2, 1)$ and $B$ is $(6, 9)$. Find the gradient $m$ of the line passing through $A$ and $B$.

A line has a gradient of $m = -1$ and passes through the point $(2, 2)$. What is its $y$-intercept?

The line $M$ shown in the diagram passes through the points $A(-4, 0)$ and $B(0, 2)$. A second line $N$ is parallel to $M$ and passes through the point $(1, 1)$. Determine the $y$-intercept of line $N$.

The line shown in the diagram passes through the points $(k, 0)$ and $(0, 2k)$, where $k > 0$. If the gradient of the line is $m$, determine the value of $m$.

The vertices of a triangle are $O(0, 0)$, $A(6, 0)$, and $B(0, 8)$. Find the equation of the line that passes through the midpoint of $OA$ and the midpoint of $OB$.

A logistics company maps its delivery route between two hubs. Hub A is located at $(-2, 3)$. The path to Hub B follows a straight line such that the perpendicular bisector of the segment $AB$ has the equation $2x - 3y + 12 = 0$. Determine the coordinates of Hub B.

A civil engineer is designing a straight access road that must pass through a supply point at $P(4, 7)$ and be perpendicular to a main highway represented by the line $3x - 4y + 12 = 0$. The new road intersects the $y$-axis at a point $Q$. Find the equation of the line representing the access road and the $y$-coordinate of point $Q$.

A mountain trail is modeled by the equation $y = 0.5x + 10$. A hiker starts at a point $H$ on the trail where $x = 4$. They decide to take a shortcut that follows a line perpendicular to the trail. If the shortcut equation is $y = ax + b$, and it passes through point $H$, find the value of the y-intercept $b$.

In the coordinate plane shown, line $L_1$ passes through the points $(0, 2)$ and $(4, 4)$. A second line, $L_2$, is perpendicular to $L_1$ and passes through the point $(2, 3)$. Find the gradient (slope) of line $L_2$.

Find the equation of the line passing through $(3, -2)$ that is parallel to the line $y = 4$.

The line $k$ has the equation $2x + 5y = 10$. What is the gradient of a line that is perpendicular to $k$?

A rectangle has vertices at $A(1, 1)$, $B(1, 4)$, and $C(5, 4)$. Given that $CD$ is parallel to $AB$, find the gradient of the line $BC$.

A line $p$ has the equation $ax + by = c$. A second line $q$ is perpendicular to $p$ and passes through the point $(b, a)$. What is the equation of line $q$?

The vertices of a triangle are $R(-2, 1)$, $S(2, 3)$, and $T(k, -1)$. If the line $RS$ is perpendicular to the line $ST$, find the value of $k$.

Line $L_1$ passes through $(a, 0)$ and $(0, b)$. Line $L_2$ is perpendicular to $L_1$ and passes through the origin. If the gradient of $L_1$ is $-\frac{3}{4}$, find the equation of $L_2$.

A path is being constructed parallel to a fence. The fence is located on the line segment connecting $(1, 2)$ and $(5, 10)$. If the path must pass through the point $(0, 0)$, find the equation of the line representing the path.

The line $L_1$ has equation $kx + 3y = 7$ and $L_2$ has equation $2x - (k-1)y = 5$. Determine the positive value of $k$ for which $L_1$ and $L_2$ are perpendicular.

The tangent to a curve at point $T(2, 4)$ is parallel to the line $y = 5x - 7$. A normal line is then drawn to the curve at the same point $T$. Find the equation of this normal line.

A circle $C$ has its center at the point $P(3, -2)$ and passes through the point $Q(7, 1)$. Find the equation of the circle $C$.

If the equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$, what does the value $r$ represent?

A circle has center $(0, 4)$ and touches the x-axis. What is its equation?

The equation of a circle is $x^2 + y^2 - 10x = 0$. Find the center of the circle.

What is the length of the chord intercepted by the circle $x^2 + y^2 = 25$ on the line $x = 3$?

The circle $x^2 + y^2 - 2x - 4y - 20 = 0$ and the line $y = x + k$ intersect at two points. Which condition on $k$ ensures this?

Find the equation of the circle that has its center on the line $y = 2$ and passes through the points $(0, 2)$ and $(4, 2)$.

A circle $C_1$ has the equation $(x+2)^2 + (y-3)^2 = 25$. A second circle $C_2$ is obtained by reflecting $C_1$ in the line $y = x$. What is the equation of $C_2$?

Find the equation of the circle which passes through the origin and makes intercepts of $6$ and $8$ on the positive $x$ and $y$ axes respectively.

A tunnel is modeled by the lower semi-circle of $(x-5)^2 + (y-4)^2 = 16$. A small horizontal pipe is installed along the line $y = 2$. Find the length of the pipe that is inside the tunnel's semi-circular cross-section.