Distance between two points, midpoint (up to 3d), angle between two lines: notes and practice questions

A map is drawn on a Cartesian plane. A straight road connects two towns, Ashton, located at A(1, 7), and Brixton, located at B(9, -1).

(a) A service station is planned to be built at the midpoint of the road connecting the two towns. Find the coordinates of the service station.

A new road, represented by the line $L$, is to be built perpendicular to the road [AB]. This new road will pass through the service station.

(b) Find the gradient of the line $L$.

(c) Hence, write down the equation of the line $L$.

A kite PQRS is shown on the following set of axes.

The kite has vertices Q(1, 5) and S(7, -1) and is symmetrical about the diagonal [PR].

(i) Write down the coordinates of the midpoint of [QS].

(ii) Hence, find the equation of the line containing the diagonal [PR].

(b) Given that vertex P lies on the y-axis and the x-coordinate of R is 6, find the area of the kite PQRS.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

A modern art sculpture features several interconnected points in 3D space. Three key connection points are $P(2, 1, 4)$, $Q(2, 7, 4)$ and $R(8, 1, 12)$.

(a) Find the length of the segment $PQ$.

(b) Find the midpoint of the segment $QR$.

(c) Find the length of the segment $PR$.

A temporary exhibition structure is being designed. The coordinates of three points on the ground are P(0,0,0), Q(8,0,0), and R(0,6,0). A supporting mast is erected at point S(4,3,10).

(a) Calculate the length of the structural cable $SQ$.

(b) Determine the angle Q$\hat{S}$R, in radians.

By considering De Moivre's theorem, show that $\cos(4\theta) = 8\cos^4\theta - 8\cos^2\theta + 1$.

Let $w = 1+i$. Show that $w^4 = -4$.

Hence, find the four roots of the equation $z^4 = -4$ in Cartesian form.

The four roots are represented by points A, B, C, D on an Argand diagram, forming a square. Find the area of this square.

Each of the points A, B, C, D is rotated counter-clockwise about the origin by an angle of $\frac{\pi}{6}$ to form new points A', B', C', D'. These points are the roots of an equation $z^4 = K$. Find $K$ in Cartesian form.

It is given that the eight points represented by the roots of $z^4=-4$ and $z^4=K$ are all solutions of $z^n = \alpha$ for some $\alpha \in \mathbb{C}$ and $n \in \mathbb{N}$. Find the smallest positive value of $n$.

A particle moves along a path defined by the equation $y = \frac{4}{x}$ for $x \neq 0$. The origin, $O(0,0)$, represents a fixed sensor. Let $P(x, y)$ be the position of the particle.

(a) Show that the square of the distance from the sensor to the particle, $|OP|^2$, can be expressed as $x^2 + \frac{16}{x^2}$.

(b) Find the coordinates of the points on the path that are closest to the sensor.

A function is defined by $f(x) = \frac{1}{2}x^2 + x + 4$. The following diagram shows part of the graph of $f$.

The graph has a vertex at V and intersects the y-axis at point P.

(a) Find the coordinates of the vertex V.

(b) Write down the coordinates of the y-intercept, P.

(c) The line L is the normal to the graph of $f$ at point P. Find the equation of L, giving your answer in the form $y=mx+c$.

(d) The line L intersects the graph of $f$ at a second point, Q. Calculate the distance between P and Q.

A plan for a garden is drawn on a coordinate grid, where 1 unit represents 1 metre. The garden consists of a paved patio and a flower bed.

The patio is a quadrilateral with vertices A(-4, 2), B(0, 5), C(4, 2), and D(0, -1).

(a) Find the area of the patio ABCD.

The flower bed is a triangle with vertices C(4, 2), E(9, 4), and F(9, 0).

(b) Find the area of the flower bed CEF.

(c) Hence, find the total area of the garden.

A data server rack is shaped like a rectangular prism. One corner of the rack is placed at the origin $O(0,0,0)$ of a 3D coordinate system. The opposite corner, $P$, has coordinates $(2, 4, 4)$ metres. The faces of the rack are parallel to the coordinate planes.

(a) Calculate the length of the main diagonal of the server rack.

(b) Determine the coordinates of the midpoint $M$ of the segment connecting vertex $C(0,0,4)$ to vertex $D(2,4,0)$.

A laser beam is emitted from a source at point $A(3, -1, 5)$. The beam reflects off a flat mirror which lies on the plane $\pi: x - 2y + 2z = 6$. The reflected beam appears to originate from a virtual source at point $B$, where $B$ is the reflection of $A$ in the plane $\pi$.

Determine the coordinates of $B$.

Find the exact distance between the laser source $A$ and the virtual source $B$.

(a) The main antenna of a communication tower is located at point $A(2, 8, 10)$. The centre of the rectangular base of the tower is at point $C(0, 5, 0)$. A vertical support beam connects the antenna to the centre of the base. Calculate the length of this support beam, $AC$.

(b) The rectangular base of the tower has dimensions $6$ m by $4$ m. Calculate the length of the diagonal of this base.

(c) A support cable runs from the antenna $A$ to one of the corners of the base, say point $P$. Find the size of the angle that this support cable $AP$ makes with the base platform.

Let P(1, 0, 1), Q(1, 2, 0), and R(k+1, 1, -1) be three points in $\mathbb{R}^3$, where k > 0.

Let $Π$ be the plane containing the points P, Q, and R.

(a) Find a Cartesian equation for the plane $Π$ in terms of k.

(b) Let N be the midpoint of the line segment [PR]. A line L passes through N and is perpendicular to the plane $Π$. Find a vector equation for the line L in terms of k.

(c) Let $L'$ be the line defined by the equations $y=x, z=1$. Show that the line L does not intersect the line $L'$ for any k > 0.