Exercises: Conic sections and loci

Find the equation of the locus of points in the plane that are equidistant from points $A(2, -3)$ and $B(-4, 5)$. What geometric shape 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 from the origin of coordinates is always equal to $5$.

Find the locus of points $P(x,y)$ whose difference of squared distances to 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 cut by a plane that does not pass through its vertex. If the plane is parallel to a single generator 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 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 equidistant from the parallel lines $r: x + 2y - 4 = 0$ and $s: x + 2y + 6 = 0$.

Marcos has to find the equation of the locus of points $P(x,y)$ whose difference of the squares of their distances to the points $A(-1, 3)$ and $B(3, 3)$ is equal to $16$, that is, $d(P,A)^2 - d(P,B)^2 = 16$.

Without performing any algebraic calculation, Marcos writes on his exam:

"Since the conic section called a hyperbola is defined from a subtraction of distances to two fixed points, this locus is necessarily a hyperbola centered at the midpoint $(1, 3)$".

1. a) Explain Marcos's conceptual error in detail: what does he confuse in the metric definition of the hyperbola, and why does the presence of squared distances completely change the type of locus obtained?
2. b) Correct the exercise: set up the equality using the generic coordinates of $P(x,y)$, expand the squares, simplify the expression, and describe the resulting figure with geometric precision (indicating its type and its relative position with respect to segment $AB$).

We want to determine the equation of the locus of points $P(x,y)$ such that the sum of the squares of their distances to the points $A(-2, 1)$ and $B(4, 5)$ is equal to $50$, that is, $d(P,A)^2 + d(P,B)^2 = 50$.

To solve it, two students discuss which strategy to follow:

• Analytical method (Lucía): substitute $P(x,y)$, expand the four squared binomials, collect all terms into the general form $x^2 + y^2 + D x + E y + F = 0$, and finally complete squares to identify the elements of the conic section.

• Synthetic-geometric method (Hugo): rely on the theoretical property that this locus is a circle whose center $C$ coincides with the midpoint of $AB$; calculate the coordinates of $C$ and determine the radius by evaluating the metric condition at an auxiliary point or by applying Apollonius's theorem (the median theorem).

1. a) Fully work out the solution applying Lucía's analytical method until obtaining the standard equation $(x-x_0)^2 + (y-y_0)^2 = R^2$.
2. b) Apply Hugo's geometric reasoning to find the center $C$ from $A$ and $B$, and verify that it coincides with the one obtained in the previous part.
3. c) Compare both approaches: which one is faster and which one carries a lower risk of algebraic calculation errors? Justify which one you would recommend on an exam.

In a telecommunications project, two transmitting antennas are located at points $A(-3,0)$ and $B(3,0)$ on a Cartesian plane (in km). A receiver $R(x,y)$ moves such 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 section it is, stating its centre and radius.

The reflector of a lighthouse has a profile designed such that the distance from any point on its surface to the light-emitting focus, situated at $F(0,2)$, is exactly equal to its distance from 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 points $(-3,0)$ and $(3,0)$ (measurements in metres). Then, they tie the ends of a rope of length $10\text{ m}$ to both stakes and, keeping the rope taut with a stylus, trace the outline on the ground. Find the equation of the locus of the edge of the flowerbed.

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

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 km). The delay in receiving the signal indicates that the ship is always $60\text{ km}$ 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 $s: x - 2y + 3 = 0$. It is desired to build a town square at a point equidistant from both avenues and which, due to planning restrictions, is situated 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 such 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 path described by the end of the arm and identify what type of conic section it represents.

Three seismic sensors are located at coordinates $A(0,0)$, $B(4,0)$ and $C(0,4)$. A calibration drone flies in the same plane such 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 path and describe the resulting shape.

During an archaeological excavation, a fragment of the rim of an ancient circular plate is discovered. To reconstruct its original size, the fragment is digitised and three points on its contour 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 original centre of the plate and its radius.

Near the Strait of Gibraltar, the Maritime Rescue center monitors the position of a disabled vessel $P(x,y)$. The coastal radar station is located at the fixed point $F(0, 3)$ (coordinates in nautical miles) and a straight maritime beaconing line follows the equation $r: y = 12$. Navigation systems confirm that the distance from the vessel to the radar station is at all times half its distance to the beaconing line.

1. a) Identify, with reasoning, what type of conic section the vessel describes (ellipse, parabola, or hyperbola), determining the numerical value of its eccentricity $e$ from the metric definition with respect to focus and directrix.
2. b) Derive the Cartesian equation of the trajectory by expanding the metric condition $\dfrac{d(P,F)}{d(P,r)} = e$ using the coordinates of $P(x,y)$.
3. c) Write the obtained equation in its standard reduced form, state the coordinates of its center, and calculate the minimum distance at which the vessel will pass from the beaconing line $y = 12$.

Consider the fixed points $A(1, 2)$ and $B(7, 2)$ in the Cartesian plane. We seek to determine the locus of points $P(x,y)$ whose sum of squared distances satisfies $d(P,A)^2 + d(P,B)^2 = 36$.

1. a) Estimate: before performing algebraic operations:

• Reason at which exact coordinates the center of the expected conic section must be located.

• Calculate the distance between $A$ and $B$. If you evaluate the distance condition directly at point $A$, what value do you obtain? Deduce whether points $A$ and $B$ themselves belong to the curve and what the value of the radius must then be.

1. b) Verify: set up the condition analytically by substituting the coordinates of $P(x,y)$, expand the squared binomials, and complete squares to obtain the standard equation $(x-x_0)^2 + (y-y_0)^2 = R^2$.
2. c) Conclude: compare the center and radius calculated in b) with your estimates from a). Explain with geometric arguments why the midpoint of segment $AB$ does not belong to this locus despite being the center of the circle.

A moving object travels in the Cartesian plane such that its distance from the vertical line $x = k$ is equal to its distance from 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 points $P(x,y)$ whose distance from the origin $O(0,0)$ is equal to $e$ times their distance from 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 along perpendicular axes that coincide with the coordinate axes. The locus of points where the product of the distances to both axes is constant and equal to $16$ defines isophonic curves. 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 segment $AB$, with endpoints at $A(-4,0)$ and $B(4,0)$. For the camera's sensors to function optimally, the viewing angle $\angle APB$ must be exactly $90^\circ$. Find analytically the locus of the possible positions of the camera, excluding points $A$ and $B$. Describe the resulting shape.

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 must pass through a fixed sensor located at $P(0,2)$. Find the equation of the locus described by the possible centres $C(x,y)$ of this cross-section and identify the conic section.

In an articulated mechanism, two rods rotate around fixed pivots $A(-a,0)$ and $B(a,0)$. The mechanism is designed such that the product of the gradients 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 point $P$ is an ellipse and determine the length of its semi-axes in terms of $a$ and $k$.

A laser beam leaves focus $F_1(c,0)$ of a reflective 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 travelled 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 sailboat navigates such that the difference of the squares of its distances to two buoys 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 sailboat's trajectory and prove analytically that this trajectory is a line perpendicular to the segment joining the two buoys.

In the vehicle dynamics department of a technology center, a test track is being designed. The layout follows a conic section defined as the locus of points $P(x,y)$ whose distance to a sensor located at $F(4, 0)$ is proportional to their distance to a vertical straight safety barrier $r: x = x_0$, satisfying the focal relationship $\dfrac{d(P,F)}{d(P,r)} = e$.

Topographical constraints require the trajectory to intersect the x-axis at exactly two points: $V_1(2, 0)$ and $V_2(-10, 0)$.

1. a) Set up the system of two equations relating the eccentricity $e$ and the position of the directrix $x_0$ using points $V_1$ and $V_2$. Solve it and discard the geometrically invalid solution (the one where the directrix contains the focus), obtaining the exact values of $e$ and $x_0$.
2. b) Classify the conic section with justification based on the value of $e$. Analytically expand the condition $\dfrac{d(P,F)}{d(P,r)} = e$ to obtain the Cartesian equation of the curve in reduced form.
3. c) From the reduced equation, calculate the coordinates of the center of symmetry of the conic section and the coordinates of its second focus $F'$.

Given the fixed points $A(0, 0)$ and $B(6, 0)$ in the Cartesian plane and a real parameter $k$, the family of loci $\mathcal{L}_k$ is defined by the condition $d(P,A)^2 + d(P,B)^2 = k$.

1. a) Expand the analytical equation of $\mathcal{L}_k$ in terms of $k$ and determine the minimum value $k_{\min}$ for which the locus is not the empty set. Interpret what degenerate geometric figure the case $k = k_{\min}$ represents.
2. b) For $k > k_{\min}$, $\mathcal{L}_k$ is a circle. Express its radius $R(k)$ in terms of the parameter $k$ and find the exact value of $k$ for which this circle is tangent to the y-axis $OY$.
3. c) For the value of $k$ obtained in part b), consider the family of lines $\mathcal{R}_m$ given by the locus $d(P,A)^2 - d(P,B)^2 = m$. Without setting up any algebraic system of equations between the line and the circle, use exclusively the geometric properties of both loci to deduce the only two values of $m$ that make $\mathcal{R}_m$ tangent to $\mathcal{L}_k$, indicating the coordinates of the respective points of tangency.