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

A circle has its center 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.

Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometers). 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 epicenter 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 center 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 center coincides with the center 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 center 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.

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 center, 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.