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SL 2.1—Equations of straight lines, parallel and perpendicular - IB Questionbank | RevisionDojo

Practice SL 2.1—Equations of straight lines, parallel and perpendicular with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A line is given by the equation $3 x+4 y=k$ , where $k \in \mathbb{R}$ . The line is tangent to a circle with centre at $(2,5)$ and radius 6 .

Find the possible value(s) of $k$ for which the line is tangent to the circle.

[5]

Question 2

SLPaper 1

Consider the points $A(-1,4)$ and $B(3,8)$ .

Find the gradient of the line segment $[A B]$ .

[2]

Find the coordinates of the midpoint $M$ of $[A B]$ .

[1]

Find the equation of the line $L$ , which is perpendicular to $[A B]$ and passes through $M$ , in the form $y=m x+c$ .

[2]

Write down the x-intercept of $L$ .

[1]

Question 3

SLPaper 1

Consider the line $L$ given by the equation $y=2 x+4$ .

Write down:

the gradient of the line,
the y-intercept,
the x-intercept.

[3]

Draw the line $L$ on the diagram below.

Figure 1: Graph for sketching $f$ and $g$

[3]

Verify whether the points $A(7,18)$ and $B(8,20)$ lie on the line.

[3]

Question 4

SLPaper 1

Consider the points $A(1,10), B(5,2)$ , and $C(-3,8)$ . The line $L$ passes through the point $A$ and is perpendicular to $BC$ .

Find the equation of $L$ .

[3]

The line $L$ passes through the point $(k, 4)$ . Find the value of $k$ .

[2]

Question 5

SLPaper 1

Consider the line $L_{1}$ given by the equation $y=2 x-3$ . Let point $A(1,9)$ lie off the line $L_{1}$ .

Find the equation of the line $L_{2}$ , which is parallel to $L_{1}$ and passes through point $A$ .

[3]

Find the equation of the line $L_{3}$ , which is perpendicular to $L_{1}$ and also passes through point $A$ .

[3]

The lines $L_{1}$ and $L_{3}$ intersect at point $B$ . Find the coordinates of point $B$ .

[2]

Determine the distance between points $A$ and $B$ .

[2]

Using your result, or otherwise, deduce the perpendicular distance from point $A$ to the line $L_{1}$ . You may support your answer with a clearly labelled sketch (not necessarily to scale).

[2]

Question 6

SLPaper 1

Let $A(3 k, 4 k)$ be a point on the line $L$ given by the equation $y=\frac{4}{3} x$ , where $k \in \mathbb{Z}$ .

Verify that the point $A$ lies on the line $L$ .

[2]

Find the possible values of $k$ such that the distance from the origin to point $A$ is equal to 10.

[4]

Write down the coordinates of the two points on the line $L$ whose distance from the origin is 10 .

[2]

Question 7

SLPaper 1

Consider the points $A(3,-5), B(-1,7)$ , and $C(5,1)$ . The line $L_{1}$ passes through point $A$ and is parallel to the line segment $B C$ .

Find the equation of $L_{1}$ .

[3]

The line $L_{2}$ passes through point $B$ and is perpendicular to $B C$ . Find the value of $k$ such that $L_{2}$ intersects the x -axis at $(k, 0)$ .

[3]

Question 8

SLPaper 1

Let $A(-2,5)$ and $B(4,9)$ be two points in the Cartesian plane.

Calculate the gradient of the line segment $[A B]$ .

[2]

Find the coordinates of the midpoint $M$ of the line segment $[A B]$ .

[1]

Determine the equation of the line that is perpendicular to $[A B]$ and passes through point $M$ .

[2]

Calculate the distance between points $A$ and $M$ .

[2]

Question 9

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Question 10

SLPaper 2

A line through $A(-4,3)$ with gradient $m$ meets $C:\ y=x^2+8x$ at $P,Q$ . The distance $PQ=11$ .

Show the $x$ -coordinates satisfy $x^2+(8-m)x+(-4m-3)=0.$

[4]

State the condition on $m$ for two distinct intersections.

[2]

Show $(x_P-x_Q)^2=m^2+76$ .

[3]

Solve for $m$ using $PQ=11$ .

[4]

SL 2.1—Equations of straight lines, parallel and perpendicular Questionbank