Straight Lines

The slope of the line joining $(0, -4)$ and $(8, 0)$ is:

Find the value of $m$ if the line through $(2, m)$ and $(4, 8)$ has a slope of $2$.

If the slope of a line is undefined, the line must be parallel to:

What is the slope of a line which makes an angle of $150^\circ$ with the positive X-axis?

The angle between the lines $y = 0$ and $y = x$ is:

If the slope of the line joining $(k, 3)$ and $(4, 5)$ is $\frac{1}{2}$, find $k$.

The slope of a line $L$ is $m$. If another line is reflected in the x-axis, the slope of the reflected line is:

If the slope of the line joining $(x, 3)$ and $(4, 8)$ is $5/2$, and the slope of the line joining $(x, 3)$ and $(8, y)$ is $1/4$, find $y$.

Find the slope of a line that makes an angle $\theta$ with the positive $x$-axis, where $\sin \theta = 3/5$ and $\theta$ is in the second quadrant.

A line $L$ passes through $(\sqrt{3}, 1)$ and is perpendicular to the line $x\sqrt{3} + y = 1$. If $L$ intersects the $x$-axis at $A$, find the slope of the line joining $A$ and $(0, 2)$.

Which of these points lies on the line $y = 2x + 1$?

Equation of a line with $x$-intercept $3$ and $y$-intercept $3$ is:

Find the slope of a line which makes an angle of $60^\circ$ with the positive direction of the $x$-axis.

A line passes through $(2, 5)$ and has a slope of $1$. Its equation is:

Find the equation of the line through $(4, -1)$ and parallel to $2x - y + 5 = 0$.

Find the equation of the line whose intercepts on the axes sum to $0$ and which passes through the point $(2, 3)$.

A line passes through $(1, 4)$ and $(3, 0)$. Its equation is:

If $p_1$ and $p_2$ are the lengths of the perpendiculars from the origin to the lines $x\sec \theta + y\csc \theta = a$ and $x\cos \theta - y\sin \theta = a\cos 2\theta$ respectively, then $4p_1^2 + p_2^2$ is equal to:

The equations of the lines through $(-1, -1)$ which are at a distance of $1$ unit from the origin are:

A line through $(1, 2)$ cuts the axes at $A$ and $B$. If the area of $\Delta OAB$ is minimum, the equation of the line is:

Reduce the equation $x - y + 4 = 0$ to normal form and find the value of $\omega$.

Find the value of $\omega$ for the line $\sqrt{3}x - y = 10$ when reduced to normal form.

Find the equation of the line where the foot of the perpendicular from the origin is $(1, 1)$.

If the foot of the perpendicular from the origin to a line is $(-2, 2\sqrt{3})$, what is the value of $p$?

Two lines $x \cos \omega_1 + y \sin \omega_1 = p_1$ and $x \cos \omega_2 + y \sin \omega_2 = p_2$ are parallel if and only if:

Find the area of the triangle formed by the coordinate axes and the line $x \cos 60^\circ + y \sin 60^\circ = 2$.

A line has $p = 6$ and $\omega = 150^\circ$. What is its x-intercept?

If $x \cos \alpha + y \sin \alpha = p$ and $x \sin \alpha - y \cos \alpha = p$ are two lines, then the distance of their point of intersection from the origin is:

The normal form of the line $x \sqrt{3} + y + 2 = 0$ is $x \cos \alpha + y \sin \alpha = p$. The values of $\alpha$ and $p$ are:

For what value of $p$ does the line $x \cos \alpha + y \sin \alpha = p$ pass through the origin?

What is the $y$-intercept of the line $x + 2y = 0$?

What is the slope of the line $x - y = 7$?

The equation $x + y + \sqrt{2} = 0$ in normal form has $p$ equal to:

The area of the triangle formed by the line $x + y = 2$ and the coordinate axes is:

The line $x \cos \alpha + y \sin \alpha = p$ is reduced to normal form. If $\alpha = 240^\circ$ and $p = 2$, what is the general equation of the line?

If the origin is shifted to $(1, 1)$, what is the new equation of the line $x + y = 2$ in the new coordinate system $(X, Y)$?

Calculate the distance between the parallel lines $3x + 4y = 5$ and $6x + 8y = 20$.

If the points $(k, 2-2k)$, $(1-k, 2k)$, and $(-k-4, 6-2k)$ are collinear, then $k$ is:

The area of the triangle formed by the line $x/a + y/b = 1$ with the coordinate axes is:

If $1/a^2 + 1/b^2 = 1/c^2$, then the line $x/a + y/b = 1$ is always at a constant distance from the origin. This distance is:

Find the distance of the point $(3, 4)$ from the line $3x + 4y = 0$.

Find the distance of the point $(2, 2)$ from the line $x + y - 4 = 0$.

What is the distance of the point $(1, 1)$ from the line $x - y + 5 = 0$?

Calculate the distance between the parallel lines $6x + 8y - 15 = 0$ and $3x + 4y + 5 = 0$.

What is the distance between the lines $4x + 3y - 11 = 0$ and $8x + 6y - 15 = 0$?

The length of the perpendicular from the origin to the line $\sqrt{3}x - y + 8 = 0$ is:

If the distance from $(1, 2)$ to the line $x\cos\theta + y\sin\theta = p$ is zero, then $p$ equals:

Consider the triangle formed by the intersection of the lines $3x - 4y = 0$, $5x + 12y = 0$, and $y = 15$. Find the distance between the origin and the in-center of this triangle.

The distance of the point $(x_1, y_1)$ from the line $ax + by + c = 0$ is $d$. If the line is rotated about the point $(x_1, y_1)$ by $90^\circ$, the distance of $(x_1, y_1)$ from the new line is:

Two sides of a square lie on the lines $x + y = 1$ and $x + y = -2$. The area of the square is:

What is the slope of a line parallel to the line $x - 7y + 5 = 0$?

Which of the following is the equation of a line parallel to the y-axis and passing through the point $(3, -2)$?

Find the slope of a line which is perpendicular to the line joining the points $(2, 3)$ and $(4, -1)$.

Two lines $L_1$ and $L_2$ have slopes $m_1$ and $m_2$ respectively. If $L_1$ is parallel to $L_2$, what is the relationship between their slopes?

The lines $x + (k-1)y + 1 = 0$ and $2x + k^2y - 1 = 0$ are perpendicular. Which of the following is a possible value of $k$?

Find the equation of the line passing through $(-1, 2)$ and parallel to the line $x - 2y + 3 = 0$.

The angle between two lines is $45^{\circ}$ and the slope of one of the lines is $\frac{1}{2}$. What is the slope of the other line?

In a city planning layout, Main Street is modeled by the line $3x - 4y + 12 = 0$. A new service road is being planned such that it is parallel to Main Street and passes through the point $(2, 3)$. Another pedestrian path is to be built perpendicular to this new service road, passing through the origin $(0, 0)$. What is the equation of the line representing the pedestrian path?

A structural engineer is designing a steel framework where two support beams must be perfectly perpendicular to ensure stability. The first beam lies along the line represented by the equation $(k-2)x + (k+3)y - 5 = 0$. The second beam is parallel to a reference line passing through the points $A(2, 5)$ and $B(4, -1)$. If the two support beams are perpendicular to each other, find the value of $k$.

A surveyor identifies two paths on a plot. Path A follows the line joining $A(h, 3)$ and $B(4, 1)$. Path B follows the line $x - 2y + 5 = 0$. If Path A is perpendicular to Path B, find the value of $h$.

What is the acute angle between the lines $x - y = 3$ and $x + y = 5$?

If two lines are parallel, what is the angle between them?

The angle between the lines $x - y = 0$ and $x + y = 0$ is:

If the slope of line $AB$ is $m$ and the slope of line $BC$ is $1/m$, what is the angle between $AB$ and $BC$ when $m=1$?

Find the tangent of the angle between the lines joining the points $(0,0), (2,3)$ and $(2,-2), (3,5)$.

The angle between the lines $2x - y + 3 = 0$ and $x + 2y - 1 = 0$ is $\alpha$. Determine the value of $\cos \alpha$.

If the angle between two lines is $\frac{\pi}{4}$ and the slope of one of the lines is $\frac{1}{2}$, find the possible slope of the other line.

A surveyor is mapping out a property boundary. One boundary line passes through the points $A(2, 3)$ and $B(4, 7)$. Another boundary line is defined by the equation $3x - y + 5 = 0$. The surveyor needs to find the tangent of the acute angle $\theta$ between these two boundary lines to calculate land area. What is $\tan \theta$?

An architect is designing a triangular park. Two sides of the park lie along the lines $x \cos 30^\circ + y \sin 30^\circ = 2$ and $x \cos 120^\circ + y \sin 120^\circ = 5$. What is the measure of the interior angle at the vertex where these two sides meet?

A laser beam is reflected off a flat mirror. The incident ray travels along the line $2x - y + 3 = 0$ and hits the mirror surface positioned along the line $x - 3y + 2 = 0$. A physicist needs to find the acute angle $\theta$ between the incident ray and the mirror to calculate the reflection properties. Determine $\theta$.

Find the distance between the two parallel lines $3x - 4y + 7 = 0$ and $3x - 4y + 5 = 0$.

Distance between $x - y = 0$ and $x - y + 5 = 0$ is:

What is the distance between the lines $x\cos\alpha + y\sin\alpha = p_1$ and $x\cos\alpha + y\sin\alpha = p_2$?

Distance between $y = 7$ and $y = -2$ is:

Find the distance between the two parallel lines represented by the equations $3x - 4y + 7 = 0$ and $3x - 4y + 2 = 0$.

Determine the distance between the parallel lines $ax + by + c = 0$ and $ax + by + c' = 0$ if the lines pass through the points $(1, 2)$ and $(1, 5)$ respectively and have a slope of $0$.

Find the distance between the parallel lines $l(x + y) + p = 0$ and $l(x + y) - r = 0$.

A civil engineer is designing a straight highway bypass and an existing railway track that can be modeled as parallel straight lines. On a coordinate map, the railway track follows the line $3x - 4y + 12 = 0$. The bypass is to be constructed such that it passes through the point $(2, -1)$ and remains parallel to the track. Calculate the perpendicular distance between the railway track and the new bypass to determine the safety buffer zone required.

A civil engineer is designing a straight highway bypass and an adjacent service road. On a coordinate mapping system, the central axis of the highway follows the line $5x - 12y + 26 = 0$ and the central axis of the service road follows the line $10x - 24y - 15 = 0$. If the engineer needs to install a drainage pipe perpendicularly connecting these two roads, what is the exact length of the pipe required? (All units are in kilometers).

In a coordinate geometry problem, two lines are given by $L_1: 7x + 24y = 50$ and $L_2: 7x + 24y = -25$. If a point $P(h, k)$ is equidistant from both lines, find the distance of $P$ from either line $L_1$ or $L_2$.