Equation of a Straight Line (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Find the equation of the line with gradient $-2$ that passes through the point $(7,-3)$, giving your answer in the form $y=mx+c$.

Find the equation of the line that passes through the points $(-2,3)$ and $(3,7)$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

Find the equation of the line that is parallel to $y=2x+3$ and passes through the point $(3,12)$.

The line $l$ passes through the points $(3,4)$ and $(9,2)$.

Find the equation of the line $l$, giving your answer in the form $y=mx+c$.

Write down the gradient of a line perpendicular to $l$.

The coordinates of the ends of the diameter of a circle are $(-3,5)$ and $(3,-3)$.

Find the length of the diameter.

Find the centre of the circle.

Three points $A$, $B$ and $C$ have coordinates $(-5,-11)$, $(1,1)$ and $(4,7)$ respectively.

Find the gradient of the line segment $AB$.

Find the gradient of the line segment $BC$.

Explain why your answers to parts (a) and (b) show that $A$, $B$ and $C$ are collinear.

A dog breeder is measuring how quickly a puppy grows by recording its "back-length", the distance from the base of the neck to the base of the tail.

At $3$ weeks old the puppy measured $6$ cm, and at $6$ weeks old it measured $8$ cm.

Find an equation, in the form $L=mw+c$, linking $L$, the back-length of the puppy in centimetres, to $w$, its age in weeks. $m$ and $c$ are constants to be found.

The equation found in part (a) is used as a model for the puppy's growth. How many centimetres per week does the model suggest the puppy grows?

Use the model to find the length of the puppy after 15 weeks.

Use the model to find the age of a puppy that has a back-length of 17 cm.

The line $l$ has equation $2x-y+3=0$.

The line $l$ crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$.

Find the coordinates of $A$ and $B$.

Find the area of triangle $OAB$, where $O$ is the origin.

An electrician charges a fixed fee of £50 plus £20 per hour.

Using $h$ for the number of hours a job takes and $P$ for the total cost of a job in pounds, write down an equation connecting $h$ and $P$.

The electrician quotes a customer a price of £200 to complete a job. How long is the electrician expecting the job to take?

A rival electrician charges a fixed fee of £38 plus £24 per hour. Write down an equation for the total cost of a job from the rival electrician.

Determine which electrician would be the cheapest for a job taking 4 hours.

A line passing through the origin $O$ is perpendicular to the line with equation $x+y=16$. The two lines meet at the point $R$. The point $P$ lies on $OR$ such that $OP:PR=3:1$.

Find the equation of the perpendicular line and hence the coordinates of $R$.

Find the coordinates of $P$.

The lines $l_1$ and $l_2$ are both parallel to the line with equation $3x-y+4=0$. The line $l_1$ passes through the origin, and $l_2$ passes through the point $(-4,7)$.

Find the equations of $l_1$ and $l_2$, giving your answers in the form $y=mx+c$.

The line $l$ passes through the points $(4,2)$ and $(8,5)$.

Find the equation of the line $l$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

The line segment $AB$ is the diameter of a circle. $A$ has coordinates $(-7,-9)$ and $B$ has coordinates $(9,3)$.

Find the coordinates of the centre of the circle and the length of the diameter.

Three points $A$, $B$ and $C$ have coordinates $(-8,1)$, $(4,4)$ and $(12,6)$ respectively.

Find the gradients of the line segments $AB$ and $BC$.

Explain why your answer to part (a) shows that $A$, $B$ and $C$ are collinear.

A dog breeder is measuring how quickly a puppy grows by recording its "back-length", the distance from the base of the neck to the base of the tail.

At $1.5$ weeks old the puppy measured $2.3$ cm, and at $6.5$ weeks old it measured $6.3$ cm.

Using a linear model, find an equation linking $L$, the back-length of the puppy in centimetres, to $w$, its age in weeks.

Use the model to find the length of the puppy at age 20 weeks.

Use the model to find the age of a puppy that has a back-length of 11.9 cm.

The line $l_1$ has equation $3x-2y+4=0$ and crosses the $x$-axis at the point $A$.

The line $l_2$ has equation $y=5-x$ and crosses the $y$-axis at the point $B$.

Find the area of triangle $OAB$, where $O$ is the origin.

Two perpendicular lines $l_1$ and $l_2$ intersect at the point $P(2,5)$.

The line $l_2$ crosses the $x$-axis at the point $Q(-3,0)$.

Find an equation for the line $l_1$.

The line $l_1$ crosses the $x$-axis at the point $R$. Find the area of triangle $PQR$.

The intersections of the following four straight lines form a parallelogram.

$2y=3x+12$

$2y=8-x$

$3x-2y-12=0$

$\frac{1}{2}x+y+2=0$

Find the coordinates of all four vertices of the parallelogram.

An electrician charges a fixed fee of £45 plus £22 per hour.

Defining suitable variables, write down an equation to represent the charges made by the electrician.

A rival electrician charges a fixed fee of £36 plus £25 per hour. Determine which electrician is cheapest for a job taking 4 hours.

The line $l_1$ has equation $5x-2y+12=0$.

The line $l_2$ has equation $2x+5y+28=0$.

Determine whether the lines $l_1$ and $l_2$ are parallel, perpendicular or neither.

The lines $l_1$ and $l_2$ intersect at the point $P$. Find the coordinates of $P$.

The line $l_1$ meets the $y$-axis at the point $Q$. The point $R$ lies on the line $PQ$ such that $PR:RQ=3:1$. Find the coordinates of $R$.

Find the equation of the line perpendicular to $2x+3y-4=0$ that passes through the point $(-1,-1)$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

A line segment $AB$ is the tangent to a circle at the point $M$. The endpoints of $AB$ have coordinates $(-5,16)$ and $(5,14)$ respectively, and $M$ is the midpoint of $AB$.

The line $MN$ is a diameter of the circle, where $N$ has coordinates $(-4,-5)$.

Find the coordinates of the centre $C$ of the circle, and the area of the circle correct to 3 significant figures.

Three points $A$, $B$ and $C$ have coordinates $(-4,-16)$, $(2,5)$ and $(10,33)$ respectively.

Show that $A$, $B$ and $C$ are collinear.

The line $l_1$ has equation $3x-2y+10=0$ and crosses the $x$-axis at the point $A$.

The line $l_2$ is perpendicular to $l_1$ and crosses the $x$-axis at $(9,0)$. The line $l_2$ crosses the $y$-axis at the point $B$.

Find the area of triangle $OAB$, where $O$ is the origin.

The point $P(-5,-2)$ lies on the line $l_1$, which crosses the $x$-axis at the point $R$.

Another line $l_2$ is perpendicular to $l_1$ at the point $P$ and crosses the $x$-axis at the point $Q(-1,0)$.

Find the area of triangle $PQR$.

The line $l$ passes through the points $(p,2p)$ and $(3p,9p)$.

Find an equation for the line $l$.

The line $l$ intercepts the $y$-axis at $(0,3)$. Find the value of $p$.

A quadrilateral $ABCD$ has vertices with coordinates $A(-1,6)$, $B(-3,2)$, $C(0,-4)$ and $D(2,0)$.

Find the equation of each of the four lines that form the quadrilateral, and state the mathematical name of the quadrilateral.

Two perpendicular lines intersect at $(-4,-4)$. One of the lines also passes through the point $(0,6)$, and the other passes through the point $(0,-5.6)$.

A kite is formed by these two lines and two others. The kite has a line of symmetry along the $y$-axis.

Find the area of the kite.

The tangent to a circle passes through the points $A(-8,-1)$ and $B(16,7)$. The tangent meets the circle at the point $N$, where $AN:NB=5:3$.

The line $NM$ is a diameter of the circle. Find the equation of the line $NM$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.