Exercises: Conic sections and loci

Find the equation of the locus of points in the plane that are equidistant from the points $A(2, -3)$ and $B(-4, 5)$. What geometric figure does it represent?

Determine the equations of the loci of points that are equidistant from the intersecting lines $r: 3x - 4y + 5 = 0$ and $s: 4x + 3y - 1 = 0$.

Find the equation of the locus of points in the plane whose distance to the origin of coordinates is always equal to $5$.

Find the locus of points $P(x,y)$ whose difference of squares of distances to the points $A(1,0)$ and $B(-1,0)$ is equal to $8$, that is, $d(P,A)^2 - d(P,B)^2 = 8$.

A cone of revolution is intersected by a plane that does not pass through its vertex. If the plane is parallel to a single generatrix of the cone, what conic section is obtained? Justify your answer based on the geometric definition of conic sections.

Find the equation of the locus of points $P(x,y)$ such that the sum of the squares of their distances to the points $A(2,0)$ and $B(-2,0)$ is equal to $20$.

Find the equations of the locus of points in the plane that are at a constant distance of $3$ units from the line $4x - 3y + 2 = 0$.

Determine the locus of points that are equidistant from the parallel lines $r: x + 2y - 4 = 0$ and $s: x + 2y + 6 = 0$.

In a telecommunications project, two transmitting antennas are located at the points $A(-3,0)$ and $B(3,0)$ of a Cartesian plane (in miles). A receiver $R(x,y)$ moves in such a way that its distance to antenna $A$ is always twice its distance to antenna $B$. Find the equation of the locus described by the receiver and identify which conic it is, indicating its center and its radius.

The reflector of a maritime lighthouse has a profile designed so that the distance from any point on its surface to the light-emitting focus, located at $F(0,2)$, is exactly equal to its distance to a horizontal reference line with equation $y = -2$. Determine the equation of the locus of the reflector's profile.

To design an elliptical flowerbed in a park, a gardener drives two stakes into the ground at the points $(-3,0)$ and $(3,0)$ (measured in feet). Then, they tie the ends of a string of length $10\text{ ft}$ to both stakes and, keeping the string taut with a peg, trace the outline on the earth. Find the equation of the locus of the edge of the flowerbed.

In a hyperbolic navigation system, a ship receives simultaneous radio signals from two coastal stations located at $A(-5,0)$ and $B(5,0)$ (in tens of miles). The delay in receiving the signal indicates that the ship is always $60\text{ mi}$ closer to station $B$ than to station $A$. Obtain the equation of the locus on which the ship is sailing.

On a city map, two straight avenues follow the equations $r: 2x + y - 4 = 0$ and $s: x - 2y + 3 = 0$. A square is to be built at a point that is equidistant from both avenues and which, due to urban planning restrictions, is located on the pedestrian street $x + y = 5$. Calculate the coordinates of the possible locations for the square.

The end $P(x,y)$ of an industrial robotic arm is programmed to move in such a way that its distance to the safety line $x = -4$ is always twice its distance to the anchor point $F(-1,0)$. Determine the equation of the trajectory described by the end of the arm and identify what type of conic it represents.

Three seismic sensors are located at the coordinates $A(0,0)$, $B(4,0)$ and $C(0,4)$. A calibration drone flies in the same plane so that the sum of the squares of its distances to the three sensors remains constant and equal to $50$. Find the equation of the locus of the drone's trajectory and describe the resulting figure.

During an archaeological excavation, a fragment of the edge of an ancient circular plate is discovered. To reconstruct its original size, the fragment is digitized and three points on its outline are taken: $A(-2, 1)$, $B(2, 5)$ and $C(4, 1)$. Using the concept of locus (intersection of perpendicular bisectors), calculate the coordinates of the plate's original center and its radius.

A moving particle travels in the Cartesian plane so that its distance to the vertical line $x = k$ is equal to its distance to the fixed point $F(2,0)$. Determine the equation of the locus in terms of the parameter $k$. For what exact value of $k$ does the locus degenerate into a line instead of forming a parabola?

Prove analytically that the locus of the points $P(x,y)$ whose distance to the origin $O(0,0)$ is equal to $e$ times their distance to the line $x = d$ (with $e > 0$ and $d > 0$) represents an ellipse if $e < 1$, a parabola if $e = 1$ and a hyperbola if $e > 1$. Write the general equation obtained before classifying it.

In an acoustic study, two sound sources emit waves on perpendicular axes that coincide with the coordinate axes. The locus of the points where the product of the distances to both axes is constant and equal to $16$ defines equal-loudness contours. Find the equation of the locus and prove that it consists of two rectangular hyperbolas, calculating the exact distance between the vertices of each of them.

A security camera located at a moving point $P(x,y)$ must monitor a linear corridor defined by the line segment $AB$, with endpoints at $A(-4,0)$ and $B(4,0)$. For the camera's sensors to work optimally, the viewing angle $\angle APB$ must be exactly $90^\circ$. Find analytically the locus of the possible positions of the camera, excluding the points $A$ and $B$. Describe the resulting figure.

In the design of a fluid system, a cylindrical pipe must be installed whose cross-section can vary in size, but must always be tangent to the x-axis and pass through a fixed sensor located at $P(0,2)$. Find the equation of the locus described by the possible centers $C(x,y)$ of this cross-section and identify the conic.

In an articulated mechanism, two rods rotate around the fixed pivots $A(-a,0)$ and $B(a,0)$. The mechanism is designed so that the product of the slopes of both rods at their point of intersection $P(x,y)$ is always a negative constant $-k^2$ (with $k > 0$). Prove that the locus of the point $P$ is an ellipse and determine the length of its semi-axes in terms of $a$ and $k$.

A laser beam starts from the focus $F_1(c,0)$ of a reflecting ellipse, bounces off a point $P(x,y)$ on its surface and arrives directly at the other focus $F_2(-c,0)$. If the total distance traveled by the light beam is $10$ units and the focal distance is $6$ units, deduce the standard equation of the locus of the ellipse using its definition based on distances.

A surveying drone flies so that the difference of the squares of its distances to two emitting beacons located at $A(-3, -1)$ and $B(5, 5)$ is always constant and equal to $32$. That is, $d(P,A)^2 - d(P,B)^2 = 32$. Find the equation of the locus of the drone's trajectory and prove analytically that this trajectory is a line perpendicular to the line segment joining the two beacons.