Coordinate Geometry

A point $P$ is located on the Cartesian plane as shown in the diagram. If point $P$ has an abscissa of $-3$ and an ordinate of $2$, in which quadrant does it lie?

If a point $P(x, y)$ is such that $x > 0$ and $y < 0$, then $P$ lies in which quadrant?

The signs of abscissa and ordinate of a point in the fourth quadrant are respectively:

A point both of whose coordinates are negative will lie in:

The abscissa of all points on a line parallel to the $y$-axis is:

What is the distance between points $A(0, 6)$ and $B(0, -2)$?

A line passes through $(0, 0)$ and $(2, 2)$. Which of these points also lies on the line?

A circle is drawn with its center at the origin $O(0, 0)$. If a point $P(3, 4)$ lies on the circle, what is the radius of the circle?

The points $A(-2, 3)$, $B(2, 3)$, $C(2, -1)$, and $D(-2, -1)$ form a figure. If the area of this figure is divided into two equal parts by the x-axis, find the ratio of the area above the x-axis to the area below the x-axis.

A treasure map indicates that a chest is hidden at point $T$. $T$ is equidistant from $A(2, 0)$ and $B(6, 0)$, and it lies on the line $y = 3$. What are the coordinates of $T$?

Observe the rectangle $ABCD$ plotted on the coordinate plane. If the coordinates of $A$, $B$, and $C$ are given as shown, what are the coordinates of the fourth vertex $D$?

A point $P$ is located in the Cartesian plane as shown. If the coordinates of $P$ are $(x, y)$, identify the quadrant in which $P$ lies and the signs of its coordinates.

A horizontal line in the Cartesian plane is called the:

What is the ordinate of the point $Q(-2, -7)$?

In the given figure, triangle $ABC$ is an equilateral triangle with side length $4$ units. If the base $BC$ lies on the x-axis such that the origin is the midpoint of $BC$, find the coordinates of the vertex $A$ (assuming $A$ lies in the positive y-direction).

Points $A(2, 2)$, $B(-2, 2)$, $C(-2, -2)$, and $D(2, -2)$ are plotted on a Cartesian plane. If these points are joined in order to form a figure, calculate the perimeter of the resulting quadrilateral.

A line segment $LM$ is drawn on the Cartesian plane. Point $L$ lies on the y-axis with an ordinate of $3$, and point $M$ lies on the x-axis with an abscissa of $-4$. What is the area of the triangle formed by the origin $O$, point $L$, and point $M$?

A line passes through $A(-2, 0)$ and $B(0, 4)$. Another line is drawn parallel to $AB$ passing through $C(1, 0)$. Where does this second line intersect the $y$-axis?

A treasure is hidden at point $T$ which is the reflection of point $P(-4, 5)$ in the $x$-axis, followed by a reflection in the $y$-axis. Find the coordinates of $T$.

In a coordinate plane, three vertices of a parallelogram $ABCD$ are $A(1, 2)$, $B(4, 3)$, and $C(6, 6)$. Find the coordinates of the fourth vertex $D$.

A point $P(a, b)$ lies in the third quadrant of the Cartesian plane. If we reflect this point across the $x$-axis to get point $Q$, and then reflect point $Q$ across the $y$-axis to get point $R$, in which quadrant does the final point $R$ lie?

What is the distance between the points $S(0, 5)$ and $T(0, -3)$ along the y-axis?

In the given diagram, which point has an ordinate of $-4$ and an abscissa of $2$?

Points $P(2, 4)$, $Q(2, -2)$, and $R(8, -2)$ are vertices of a triangle. What type of triangle is $PQR$?

In which quadrant does the point $T(x, y)$ lie if $xy > 0$ and $x + y < 0$?

A circle is centered at the origin $(0,0)$ and passes through the point $(3, 4)$. What is the radius of this circle?

The point $K(k-2, 2k+1)$ lies on the y-axis. Find the value of $k$ and the coordinates of $K$.

A surveyor identifies three points of a rectangular field as $A(-2, 1)$, $B(4, 1)$, and $C(4, 5)$. Find the coordinates of the fourth vertex $D$ and the perimeter of the field.

Two points $A(2, 3)$ and $B(2, -5)$ are given. What is the length of the segment $AB$ and the coordinates of its midpoint?

If a point $S(a, b)$ lies in the third quadrant, what can be said about the coordinates of point $S'$ which is the reflection of $S$ in the origin?

Reflecting the point $(-1, -1)$ in the origin results in:

What point is invariant (stays the same) when reflected in the y-axis?

A point lies on the x-axis at a distance of 3 units from the y-axis to the right. If it is reflected in the x-axis, its new position is:

If point $C(2, 2)$ is reflected in the line $x=0$, what is the new coordinate?

A line segment $AB$ with $A(2, 3)$ and $B(5, 3)$ is reflected in the y-axis. What is the length of the image segment $A'B'$?

The point $P(a, b)$ is first reflected in the origin to $P_1$ and then $P_1$ is reflected in the x-axis to $P_2$. If $P_2$ is $(3, 4)$, find $a$ and $b$.

A square has vertices $(1, 1), (2, 1), (2, 2), (1, 2)$. If it is reflected in the line $y = -x$, one of the new vertices will be:

In a coordinate plane, point $M(3, 2)$ is reflected across the line $y = k$ to $M'(3, 8)$. What is the value of $k$?

A circle is centered at $(4, 4)$ with radius $2$. It is reflected across the line $x = 2$. What is the equation of the line passing through the centers of the original circle and its reflected image?

Consider point $P(x, y)$. If $P$ is reflected across the $x$-axis to $P_1$, and $P_1$ is then reflected across the line $y = x$ to $P_2$, which coordinates represent $P_2$?

The perpendicular distance of the point $L(2, 3)$ from the $x$-axis is $h$ and from the $y$-axis is $k$. Find $h + k$.

A point whose abscissa is $-4$ and ordinate is $0$ lies on which of the following?

What is the sum of the perpendicular distances of point $P(-3, 4)$ from the $x$-axis and the $y$-axis?

Point $T$ is $2$ units to the left of the $y$-axis and $6$ units above the $x$-axis. What is its $x$-coordinate?

The area of a triangle formed by the points $O(0, 0)$, $A(4, 0)$, and $B(0, 3)$ is found using the perpendicular distances of $A$ and $B$ from the origin. What is the area?

If the perpendicular distance of a point $P(x, y)$ from the $x$-axis is $5$ and from the $y$-axis is $2$, and the point lies in the fourth quadrant, find $x - y$.

A circle has its center at $(2, 2)$. If the circle is tangent to both axes, what is the perpendicular distance from the center to any point on the $y$-axis at the point of tangency?

A line segment $AB$ of length $10$ units has its endpoint $A$ on the $x$-axis and $B$ on the $y$-axis. If the perpendicular distance of the midpoint of $AB$ from the $y$-axis is $3$ units, find the perpendicular distance of the midpoint from the $x$-axis.

If the perpendicular distance of a point $P(x, y)$ from the $x$-axis is $h$ and from the $y$-axis is $k$, then the coordinates of $P$ in the fourth quadrant are given by:

Find the coordinates of a point which is equidistant from the $x$-axis and the $y$-axis, and its perpendicular distance from the origin is $\sqrt{50}$ units. The point lies in the third quadrant.

Calculate the slope of the line that passes through the points $(1, -3)$ and $(3, 5)$.

Two points on a line are $(2, 5)$ and $(4, 5)$. What is the slope of this line?

Find the slope of a line segment that drops 4 units for every 2 units it moves to the right.

The graph shows a line passing through $(-3, -3)$ and $(3, 3)$. What is its slope?

A line passes through $(a, 0)$ and $(0, b)$. What is the slope of this line?

In the rectangle $ABCD$ shown, $A$ is $(0,0)$ and $B$ is $(6,0)$. If the slope of $AC$ is $\frac{1}{2}$, find the coordinates of $D$.

Find the value of $m$ such that the line through $(2, 3)$ and $(5, m)$ is parallel to the line through $(1, 1)$ and $(3, 5)$.

The profit of a small company increases linearly. In year 2, the profit was $10$ lakhs. In year 5, the profit was $25$ lakhs. If the 'growth rate' is defined as the slope of the profit-year line, what is the growth rate in lakhs per year?

A ladder is leaning against a wall. The foot of the ladder is at $(3, 0)$ and it touches the wall at $(0, 4)$. A second ladder is placed parallel to the first one, but its foot is at $(6, 0)$. At what height does the second ladder touch the wall?

A square $ABCD$ is plotted on a graph. Side $AB$ lies on a line with a slope of $3$. What is the slope of the side $BC$, which is adjacent and perpendicular to $AB$?

A line has a slope of $3/4$. If the change in $x$ (run) is $8$, what is the change in $y$ (rise)?

Find the slope of a line passing through $(3, -2)$ and $(3, 5)$.

If the slope of a line is $0$, the line is:

Calculate the slope of the line segment shown between $(x_1, y_1) = (-2, -3)$ and $(x_2, y_2) = (2, 1)$.

In a coordinate plane, the slope of the line segment connecting $(2, y)$ and $(5, 8)$ is $2$. Find $y$.

A line passes through $(a, b)$ and $(c, d)$. If the line is vertical, which condition must hold?

If the slope of the line joining $A(k, 2)$ and $B(3, 4)$ is $2$, find the value of $k$.

Two laser beams are fired from different sources. Beam A passes through $(1, 2)$ and $(3, 8)$. Beam B is parallel to Beam A and passes through the point $(0, -1)$. What is the x-coordinate of the point on Beam B where the y-coordinate is $14$?

A rectangular coordinate system shows a river bank modeled by the line $y = \frac{1}{3}x + 2$. A swimmer starts at $(4, 1)$ and swims in a straight line perpendicular to the bank. What is the slope of the swimmer's path?

The growth of a plant is modeled as a linear function. At day 2, the height is $5$ cm. At day 10, the height is $25$ cm. If the growth remains constant, what is the slope of the line representing this growth, and what does it represent?

Which of these lines has no x-intercept?

Find the y-intercept of the line $3y = -2x + 12$.

What is the x-intercept of the line $x/a + y/b = 1$?

What is the x-intercept of the line $y = 2x - 4$?

A line has a y-intercept of $3$ and an x-intercept of $-5$. What is its slope?

The intercepts of a line are $a$ and $b$. If the line passes through $(a, b)$, what is the value of $a + b$?

A line segment $PQ$ with $P$ on the x-axis and $Q$ on the y-axis has a length of $10$. If the y-intercept is $6$, what is the x-intercept?

An equilateral triangle has one vertex at the origin and another vertex on the $x$-axis at $(4, 0)$. If the third vertex lies in the first quadrant, find the $y$-intercept of the line passing through the third vertex and the point $(4, 0)$.

A line segment $PQ$ has its endpoints on the axes, with $P$ on the $x$-axis and $Q$ on the $y$-axis. The midpoint of $PQ$ is $M(3, 4)$. If a circle is drawn with $PQ$ as the diameter, find the $y$-intercept of the tangent to the circle at point $Q$.

The intercept form of a line is $\frac{x}{a} + \frac{y}{b} = 1$. If this line is rotated $90^{\circ}$ clockwise about its $x$-intercept, what is the $y$-intercept of the new line?

The linear equation $3x - 2y = 12$ represents a straight line. Determine the y-intercept of this line.

In the given figure, a line segment $PQ$ is shown. If the line segment is extended to form a straight line, at what value of $y$ will it cross the y-axis?

A line passes through the points $(2, 5)$ and $(4, 9)$. What is the y-coordinate of the point where this line intersects the vertical axis?

The $y$-intercept of a line is the value of $y$ when:

A line $k$ has a y-intercept of $b$. If the line passes through $(3, 2)$ and $(6, 4)$, find the value of $b$.

The line segment joining $M(-2, 6)$ and $N(4, -3)$ is divided by the y-axis. At what y-coordinate does this segment cross the y-axis?

Two lines $L_1$ and $L_2$ intersect at the point $(2, 3)$. $L_1$ passes through the origin $(0,0)$. $L_2$ is perpendicular to the x-axis. What is the y-intercept of $L_1$?

A laser beam follows a path described by the equation $y = mx + c$. It is known that the beam passes through the point $P(2, 5)$ and the area of the right-angled triangle formed by the beam, the x-axis, and the y-axis in the first quadrant is $16$ square units. Given the beam has a negative slope, determine the y-intercept ($c$) of the laser beam's path.