Two roads in Dublin may never cross (parallel), cross at a single point (intersecting) or actually be the same road seen under another name (coincident). Two lines in the plane can only be in one of those three situations.
Three possible positions
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Write the relative position of the lines $r: y = 3x - 5$ and $s: y = -2x + 4$, justifying your answer using their gradients.
Write the relative position of the lines $r: y = 3x - 5$ and $s: y = -2x + 4$, justifying your answer using their gradients.
The way to decide the position of two lines is to compare their slopes. Write each line as $y = mx + c$ (or find $m = -\frac{a}{b}$ from $ax+by+c=0$). If $m_1 = m_2$ and the y-intercepts differ, the lines are parallel; if $m_1 = m_2$ and the y-intercepts are equal, they are coincident; if $m_1 \ne m_2$, they intersect.
To use slopes with a line in parametric form such as $x = 2 - t,\ y = 3 + 2t$, eliminate $t$: from the first equation $t = 2 - x$, so $y = 3 + 2(2 - x) = 7 - 2x$. Then $y = -2x + 7$, so the slope is $-2$ and you can compare it with another line in the usual way.
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Determine the relative position of the lines $r: 2x - 4y + 6 = 0$ and $s: -x + 2y - 3 = 0$ by comparing their coefficients.
Determine the relative position of the lines $r: 2x - 4y + 6 = 0$ and $s: -x + 2y - 3 = 0$ by comparing their coefficients.
Parallel or intersecting lines given in other forms
The same idea works for lines in any form: find the slope of each line (for $y = mx + c$ it is $m$; from two points it is rise ÷ run) and compare. To use the coefficient test, first convert to the form $ax+by+c=0$.
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Find the explicit equation of the line parallel to $r: y = -4x + 1$ passing through the point $P(0, 3)$.
Find the explicit equation of the line parallel to $r: y = -4x + 1$ passing through the point $P(0, 3)$.
Two lines are perpendicular when $m_1 m_2 = -1$. This lets us find a line through a point at right angles to another line.
Area of a triangle
If one vertex is at the origin $O(0,0)$ and the others are $A(x_1, y_1)$ and $B(x_2, y_2)$, the area is $\frac{1}{2}|x_1 y_2 - x_2 y_1|$ (you will find this in the Formulae and Tables booklet). For any triangle, translate it so that one vertex is at the origin: subtract that vertex's coordinates from the other two.
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Given the lines $r: x = 2 - t, y = 3 + 2t$ and $s: 2x + y - 7 = 0$, study their relative position by comparing their direction vectors and checking whether a point on…
Given the lines $r: x = 2 - t, y = 3 + 2t$ and $s: 2x + y - 7 = 0$, study their relative position by comparing their direction vectors and checking whether a point on $r$ belongs to $s$.
Two main irrigation pipes in a field follow the paths of the lines $r: 5x + 3y - 15 = 0$ and $s: 2x - y + 5 = 0$. Find out whether they cross and, if so, find the exact…
Two main irrigation pipes in a field follow the paths of the lines $r: 5x + 3y - 15 = 0$ and $s: 2x - y + 5 = 0$. Find out whether they cross and, if so, find the exact coordinates of the connection point.
Three streets form a triangular park in the centre of a city. The equations of the streets are $r: x + y - 6 = 0$, $s: 2x - y - 3 = 0$ and $t: x - 2y + 6 = 0$. Calculate…
Three streets form a triangular park in the centre of a city. The equations of the streets are $r: x + y - 6 = 0$, $s: 2x - y - 3 = 0$ and $t: x - 2y + 6 = 0$. Calculate the exact coordinates of the three vertices of the park.
The reflection $A'$ of a point $A$ in a line $r$ is found in two steps: draw the line through $A$ perpendicular to $r$, find where it meets $r$ (call it $M$), then $M$ is the midpoint of $[AA']$.
For a ray of light hitting a mirror, reflect a point of the incoming ray in the mirror line. The reflected ray passes through the point where the ray meets the mirror and through that image point.
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Find the coordinates of the reflection of point $A(2, 5)$ in the line $r: x - 2y + 3 = 0$.
Find the coordinates of the reflection of point $A(2, 5)$ in the line $r: x - 2y + 3 = 0$.
A ray of light follows the path of the line $r: x - y + 2 = 0$ and strikes a flat mirror located along the x-axis. Given that the angle of incidence equals the angle of…
A ray of light follows the path of the line $r: x - y + 2 = 0$ and strikes a flat mirror located along the x-axis. Given that the angle of incidence equals the angle of reflection relative to the normal, find the general equation of the line describing the reflected ray.