Lines in the Plane practice questions

Question 1

The equation of the line through $A(0, 1)$ and $B(2, 5)$ is:

Accepted Answer

Slope $= 2$. The line through $(0, 1)$: $y - 1 = 2(x - 0) \Rightarrow y = 2x + 1$.

Answer

$y = x + 5$

Answer

$y = 2x - 1$

Answer

$y = 2x + 1$

Answer

$y = -2x + 5$

Question 2

The slope of the line $y = 3x - 7$ is:

Accepted Answer

In slope-intercept form $y = mx + b$, the slope is $m = 3$.

Answer

$-7$

Answer

$-3$

Answer

$3$

Answer

$7$

Question 3

The $y$-intercept of $y = -2x + 5$ is:

Accepted Answer

The constant term in slope-intercept form is the $y$-intercept: $b = 5$.

Answer

$-5$

Answer

$-2$

Answer

$5$

Answer

$2$

Question 4

The slope of the line $y = -3x + 5$ is:

Accepted Answer

The line is in slope-intercept form $y = mx + b$ with $m = -3$ and $b = 5$.

So the slope is $-3$.

Answer

$5$

Answer

$3$

Answer

$-3$

Answer

$\dfrac{5}{3}$

Question 5

Determine $a \in \mathbb{R}$ so that $A(1, 2)$, $B(3, 4)$, and $C(a, 5)$ are collinear.

Accepted Answer

$\overrightarrow{AB} = (2, 2)$, $\overrightarrow{AC} = (a - 1, 3)$. Collinear ⇒ $\dfrac{a - 1}{2} = \dfrac{3}{2} \Rightarrow a - 1 = 3 \Rightarrow a = 4$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$5$

Question 6

The distance from $A(2, 3)$ to the line $3x - 4y + 5 = 0$ is:

Accepted Answer

$d = \dfrac{|3 \cdot 2 - 4 \cdot 3 + 5|}{\sqrt{9 + 16}} = \dfrac{|-1|}{5} = \dfrac{1}{5}$.

Answer

$\dfrac{1}{5}$

Answer

$\dfrac{2}{5}$

Answer

$\dfrac{3}{5}$

Answer

$1$

Question 7

A line parallel to $y = -3x + 2$ passing through $(1, 4)$ has equation:

Accepted Answer

Slope $= -3$. Line through $(1, 4)$: $y - 4 = -3(x - 1) \Rightarrow y = -3x + 7$.

Answer

$y = 3x + 1$

Answer

$y = -3x + 4$

Answer

$y = -3x + 7$

Answer

$y = \dfrac{1}{3}x + 4$

Question 8

A line perpendicular to $y = 2x + 1$ passing through the origin has slope:

Accepted Answer

$m_1 \cdot m_2 = -1 \Rightarrow m_2 = -\dfrac{1}{2}$.

Answer

$-2$

Answer

$-\dfrac{1}{2}$

Answer

$\dfrac{1}{2}$

Answer

$2$

Question 9

The line $2x - 3y + 6 = 0$ crosses the $x$-axis at:

Accepted Answer

$2x + 6 = 0 \Rightarrow x = -3$, so the line crosses the $x$-axis at $(-3, 0)$.

Answer

$(-3, 0)$

Answer

$(0, -3)$

Answer

$(0, 2)$

Answer

$(3, 0)$

Question 10

The lines $y = 2x + 3$ and $y = 2x - 1$ are:

Accepted Answer

Both lines have slope $2$ but different $y$-intercepts ($3$ and $-1$).

Same slope, different intercept ⇒ parallel and distinct (never meet).

Answer

parallel and distinct

Answer

the same line

Answer

perpendicular

Answer

intersecting at exactly one point

Lines in the Plane

Problems & worked solutions

Problem 1 — Lines in the Plane

Problem 2 — Lines in the Plane

Problem 3 — Lines in the Plane

Problem 4 — Equation of a Line

Problem 5 — Lines in the Plane

Problem 6 — Lines in the Plane

Problem 7 — Lines in the Plane

Problem 8 — Lines in the Plane

Problem 9 — Lines in the Plane

Problem 10 — Relative Positions of Lines

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