Exercises: Exercises

Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.

Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.

Determine the coordinates of the centre and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.

A circle has its centre at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.

Find the lengths of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.

Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.

Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.

Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its standard equation.

In an Analytic Geometry exam, Marcos comes across the following statement: "Determine the locus of points $P(x, y)$ in the plane whose distance to the point $A(-3, 0)$ is exactly three times its distance to the point $B(1, 0)$". Without performing any operations, Marcos claims: "Since there are two fixed points acting as foci and a proportionality constant, the resulting curve is definitely an ellipse with foci at $A$ and $B$".

1. a) Explain with reasons the conceptual error made by Marcos based on the rigorous metric definition of an ellipse versus the condition given in the statement.
2. b) Formulate the condition of the statement analytically ($d(P, A) = 3 \cdot d(P, B)$), eliminate the radicals, and obtain the standard or ordinary equation of the conic to show Marcos what geometric figure is actually obtained, specifying the coordinates of its centre and the value of its radius.

Design and construct, justifying the choice of each parameter, the general equation of a conic of the form $Ax^2 + By^2 + Dx + Ey + F = 0$ with integer coefficients that simultaneously satisfies the following three conditions:

1) It is an equilateral hyperbola (its asymptotes are perpendicular to each other and $a = b$).

2) Its centre is located at the point $C(2, -1)$.

3) It passes through the origin of coordinates $(0, 0)$.

Once you obtain the expanded general equation, explain in detail what relationship between the signs and values of the coefficients $A$ and $B$ guarantees that the conic is a hyperbola and why its asymptotes turn out to be perpendicular.

Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometres). Assuming that the seismic waves travel at the same speed and arrive at the same time at all three stations, find the coordinates of the epicentre and the radius of the affected area (distance to the stations).

A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is $3 \text{ Mm}$ and the maximum distance (apoapsis) is $9 \text{ Mm}$. Determine the equation of the orbit and its eccentricity, assuming that the centre of the ellipse is at the origin and the focal axis lies on the x-axis.

The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is at a height of $50 \text{ m}$ above the ground. If the base of the tower has a width of $80 \text{ m}$, find the equation of the hyperbola, assuming its centre coincides with the centre of the narrowest part.

The main cable of a suspension bridge forms a parabola. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck. The lowest point of the cable is $10 \text{ m}$ above the deck. Determine the equation of the parabola and calculate the height of the cable $30 \text{ m}$ from the centre of the bridge.

A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path given by the equation $3x + 4y + k = 0$. Determine the values of the parameter $k$ for which the ship is detected by the radar at some point along its path.

An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the equation of the drone's path, simplify it, and identify the type of geometric curve it describes.

A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has x-coordinate $x = 3$ and a positive y-coordinate. Find the equation of the tangent line to the mirror at that point.

Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of intersection.

You can do it on paper and send a photo with your answer.

Consider the ellipse with standard equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the line $r: x + y - 4 = 0$.

1. a) Before operating algebraically, make a reasoned geometric estimation: obtain the four vertices of the ellipse, determine the intersection points of the line $r$ with the coordinate axes, and deduce with justification whether the line is exterior, tangent, or secant to the ellipse, indicating whether you expect to find zero, one, or two intersection points.
2. b) Check your estimation using the analytical method: set up the system of equations, substitute the line into the ellipse, calculate the discriminant ($\Delta$) of the resulting quadratic equation, and interpret its sign. Finally, find the exact coordinates of the points of contact to confirm whether your initial prediction was correct.

Inspired by the famous "Hall of Secrets" at the Monastery of El Escorial (Madrid), a science outreach room has a semi-elliptical vault to take advantage of the focal acoustic property: any whisper emitted from one of its foci bounces off the ceiling and is heard clearly at the other focus. The horizontal base of the floor measures $10\text{ m}$ in width (major axis coinciding with the floor) and the maximum height at the centre of the room is $3\text{ m}$.

1. a) Place the origin of coordinates at the centre of the floor and write the standard equation of the ellipse that models the ceiling cross-section, identifying the semi-axes $a$ and $b$.
2. b) At what exact distance from the side walls must two visitors stand in order to be positioned over the acoustic foci and experience the phenomenon?
3. c) A vertical sculpture $2,5\text{ m}$ high is to be placed standing on the floor at a horizontal distance of $3\text{ m}$ from the centre of the room. Justify by calculating the ceiling height at that point whether the sculpture will touch the ceiling or not.

A comet follows a parabolic path given by the equation $y^2 = 8x$. An asteroid follows a rectangular hyperbolic orbit described by $x^2 - y^2 = 9$. Find the exact coordinates of the points of intersection of both paths.

From a control station located at point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation $x^2 + y^2 - 4x + 2y - 15 = 0$. Find the equations of the two straight-line paths tangent to the orbit from the control station.

Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is equal to $-\frac{9}{16}$. Identify the resulting conic, calculate its eccentricity, and sketch it.

A car headlight has a parabolic cross-section. The bulb is located at the focus $F(3, 0)$ and the vertex of the parabola is at the origin. A ray of light leaves the bulb, strikes the parabola at a point with y-coordinate $y = 6$, and is reflected. Calculate the equation of the tangent line at the point of incidence and find the angle that the incident ray makes with this tangent.

Two telecommunications networks have coverage areas modelled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of the radical axis, prove analytically that the circles intersect at two points, and calculate the exact distance between their points of intersection.

Given the conic with general equation $9x^2 + 25y^2 - 54x + 200y + 256 = 0$, classify it by completing the square, and find the coordinates of its centre, its vertices, its foci, and the value of its eccentricity.

A subatomic particle moves in a magnetic field, describing an equilateral hyperbolic path whose asymptotes coincide with the coordinate axes. Knowing that the particle passes through the point $P(4, 2)$, solve:

1. a) Find the equation of the path.

1. b) Determine the exact coordinates of the vertices and the foci.

1. c) Sketch the curve (the figure shows the axes and the point $P$; you can copy it into your notebook).

You can do it on paper and send a photo with your answer.

Find the equation of the locus of points in the plane whose distance from the fixed point $F(4, 0)$ is equal to half their distance from the vertical line $x = 16$. Identify the resulting conic, find its semi-axes, and sketch it, illustrating the directrix property.

To find the tangent lines to the circle $C: x^2 + y^2 - 2x - 4y - 4 = 0$ drawn from the exterior point $P(5, 5)$, two methodological approaches are proposed:

• Method A (General algebraic): Set up the family of lines passing through $P$, $y - 5 = m(x - 5)$, substitute into the conic, and set the discriminant of the resulting quadratic equation to zero ($\Delta = 0$).

• Method B (Metric geometric): Obtain the centre $C$ and the radius $R$ of the circle by completing squares, and set the distance from the centre to the general line $mx - y + (5 - 5m) = 0$ equal to the radius ($d(C, r) = R$).

1. a) Solve the problem by fully applying both methods to obtain the values of the slopes $m$, and write the general equations of the two resulting tangents.
2. b) Critically compare both methods: which of the two turns out to be notably more agile and with a lower computational burden in this case? Could you apply Method B with the same simplicity if the conic were an ellipse of the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$? Justify your answer based on the geometric properties of both curves.

The Maritime Rescue Centre of Finisterre (Galicia) monitors the trajectory of a drifting buoy in the ocean using two fixed stations located at $A(-5, 0)$ and $B(5, 0)$ (coordinates in kilometres).

1. a) In a first phase, the buoy $P(x, y)$ emits acoustic pulses that reach $B$ before $A$, with the difference in distances to both stations remaining constant: $d(P, A) - d(P, B) = 6\text{ km}$. Identify with reasons which type of conic and which of its two branches the buoy moves on, deduce its parameters ($a, b, c$), and write its standard equation.
2. b) In a second phase, the buoy enters the coastal area in front of a straight coastline described by $r: x = -7$. Sensors record that along that stretch, the buoy moves keeping at all times the same distance to station $A(-5, 0)$ as to the coastline $r$. Identify this second conic, determine the coordinates of its vertex, its focus, and its parameter $p$, and obtain its canonical or standard equation.
3. c) Analytically determine the coordinates of the points where both recorded trajectories could have crossed (intersection of the two conics in the region $x > 0$) and discuss the geometric consistency of the solutions obtained.

Topic challenge. Orbital rescue mission: from ellipse to hyperbola.

Your goal is to model the trajectory of a celestial body and formally analyze its conic transitions.

1. Data collection: Go to NASA's JPL Small-Body Database and select a comet or near-Earth asteroid. Record its semi-major axis $a$ (in AU, e.g., $1,45\text{ UA}$) and its eccentricity $e$.

2. Construction in GeoGebra: Define the ellipse as a locus based on its foci. Deduce its standard equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, calculating $b=\sqrt{a^2-c^2}$. Also plot the associated director circle of radius $a$.

3. Limits and continuity: Model eccentricity as a continuous function $e(c)=\frac{c}{a}$ on $[0, a]$. Justify its continuity and calculate $\lim_{c \to 0} e(c)$ to geometrically explain the limiting case of the circle. Consider what happens if a perturbation increases $e > 1$, transforming the orbit into an escape hyperbola; analytically determine its asymptotes $y = \pm \frac{b}{a}x$, redefining for the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ the parameter $b=\sqrt{c^2-a^2}$ (now $c>a$).

4. Final product: Prepare a concise report (maximum two pages) with the GeoGebra file or screenshots, the algebraic derivations, and your physical conclusions.

Assessment criteria:

• Precision in modeling conics using real astronomical data.

• Algebraic rigour in the deduction of loci, continuity, and limit calculations.

• Analytical quality and geometric justification of orbital transitions in the report.