Exercises: Ellipse

Given the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$, determine the coordinates of its vertices, those of its foci, and the value of its eccentricity.

Write the standard equation of an ellipse centred at the origin of coordinates, knowing that one of its foci is the point $F(4, 0)$ and that the length of its major axis is $10$.

Calculate the eccentricity of the ellipse whose equation is $16x^2 + 25y^2 = 400$.

An ellipse has its centre at the point $C(-2, 3)$, its major axis is parallel to the x-axis and measures $12$ units, and its minor axis measures $8$ units. Write its equation in canonical form.

Check whether the points $P(2, -3)$, $Q(5, 0)$ and $R(4, 2)$ lie inside, outside, or on the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$.

Find the equation of an ellipse centred at the origin, whose foci lie on the y-axis, knowing that the focal distance is $8$ units and the eccentricity is $0,8$.

From the equation of the ellipse $\frac{(x-1)^2}{16} + \frac{(y+4)^2}{9} = 1$, state the coordinates of its centre and the lengths of its semi-axes.

The eccentricity of an ellipse is $\frac{1}{3}$ and its semi-minor axis measures $4\sqrt{2}$. Calculate the measure of its semi-major axis.

Lucía has to determine the characteristic elements of the ellipse with equation $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$. On her exam, she writes the following solution:

"Since the first denominator is $9$, the semi-major axis is $a = \sqrt{9} = 3$. Since the second denominator is $25$, the semi-minor axis is $b = \sqrt{25} = 5$. Since $x^2$ is in the first position, the foci lie on the $X$-axis. Applying the Pythagorean theorem, the focal distance is $c = \sqrt{a^2 + b^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5{,}83$. Therefore, the foci are $F_1(-\sqrt{34}, 0)$ and $F_2(\sqrt{34}, 0)$, and the eccentricity is $e = \dfrac{c}{a} = \dfrac{\sqrt{34}}{3} \approx 1{,}94$".

1. a) Identify and explain the three conceptual errors that Lucía has made.
2. b) Using the right triangle that relates $a$, $b$, and $c$, justify why it is geometrically impossible for the eccentricity of an ellipse to be greater than $1$.
3. c) Correct the exercise by determining the correct values of the semi-axes, the exact coordinates of the foci, and the value of the eccentricity.

You want to design two different concentric ellipses, both centered at the origin and with their foci located on the $X$-axis at the same points $F_1(-4, 0)$ and $F_2(4, 0)$, but with different degrees of flattening:

• Ellipse $E_1$ must have an eccentricity $e_1 = 0{,}8$.

• Ellipse $E_2$ must have an eccentricity $e_2 = 0{,}5$.

1. a) Algebraically deduce the values of the semi-axes $a$ and $b$ for each of the two ellipses and write their corresponding canonical equations.
2. b) Without needing to plot them using a table of values, reason which of the two ellipses will reach a greater vertical width (that is, a greater length of its minor axis). Justify your answer by analyzing how the semi-minor axis $b = \sqrt{a^2 - c^2}$ varies as the eccentricity decreases when the distance between the foci remains constant.

The orbit of Mercury around the Sun is an ellipse with the Sun at one of its foci. The minimum distance from Mercury to the Sun (perihelion) is $46$ million kilometres, and the maximum distance (aphelion) is $70$ million kilometres. Calculate the eccentricity of Mercury's orbit and the length of its semi-minor axis in million kilometres.

A stone bridge over a river has the shape of a semi-elliptical arch. The width of the river (major axis of the ellipse) is $20\text{ m}$ and the maximum height of the arch at the centre of the river is $6\text{ m}$. Calculate the height of the arch at a distance of $4\text{ m}$ from the centre of the river.

Given the general equation $9x^2 + 4y^2 - 36x + 24y + 36 = 0$, prove that it corresponds to an ellipse. To do so, express it in canonical form and determine the coordinates of its centre, its vertices, and its foci.

Find the equation of the locus of points in the plane $P(x, y)$ whose sum of distances to the points $F_1(-3, 0)$ and $F_2(3, 0)$ is always equal to $10$. Simplify the expression to obtain the standard equation of an ellipse.

In a whispering gallery, which has the shape of an ellipse, two people located at the foci can hear each other perfectly even if they speak in a whisper. If the gallery has a total length of $34\text{ m}$ and a maximum width of $16\text{ m}$, at what distance from the centre of the room should each person stand to take advantage of this acoustic phenomenon?

Study the relative position of the line with equation $2x + y - 5 = 0$ and the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. If they intersect, find the coordinates of the points of intersection.

Determine the locus of points in the plane whose distance to the point $F(4, 0)$ is equal to half their distance to the line with equation $x - 16 = 0$. Identify the resulting conic section and calculate its centre and the lengths of its semi-axes.

In a clockwork mechanism, a gear has an elliptical shape. It is known that its boundary equation is given by $x^2 + 4y^2 = 16$. A small pin is to be placed at a point on the ellipse whose x-coordinate is twice its y-coordinate. Calculate the coordinates of this point (assuming both are positive) and the exact distance from this point to the centre of the gear.

The perimeter track of a sports hall has an elliptical shape with its major axis lying on the $X$-axis. The maximum length of the venue (major axis) is $100\text{ m}$ ($a = 50\text{ m}$) and the maximum width (minor axis) is $96\text{ m}$ ($b = 48\text{ m}$).

1. a) Estimate: Notice that the semi-axes only differ by $4\,\%$ ($48\text{ m}$ compared to $50\text{ m}$). Before calculating anything, intuitively estimate how far from the center you think the foci will be (less than $5\text{ m}$, or more than $10\text{ m}$?) and whether the eccentricity will be below $0{,}1$ or noticeably above it. Briefly justify your intuition.
2. b) Calculate: Find the exact value of the focal distance $c$, the coordinates of the foci, and the eccentricity $e$.
3. c) Check and reflect: Compare the exact result obtained with your initial estimate. Are the foci as close to the center as the small difference between the semi-axes seemed to suggest? Explain why the Pythagorean relationship $c = \sqrt{a^2 - b^2}$ causes small percentage variations between $a$ and $b$ to produce focal displacements and eccentricity values that are unexpectedly large to the human eye.

A tunnel whose cross-section is shaped like a semi-ellipse (cut along its major axis on the horizontal pavement) is planned for a regional road. The total usable width of the roadway at the base is $10\text{ m}$ and the maximum height at the central point of the vault is $4\text{ m}$. The roadway has two traffic lanes of $3\text{ m}$ in width each (one per direction), positioned symmetrically with respect to the center, and flanked by $1\text{ m}$ wide shoulders on the outer ends.

1. a) Place the origin of coordinates at the center of the roadway on the ground and write the mathematical equation describing the profile of the tunnel vault, indicating the range of validity of the variables.
2. b) A special transport vehicle with a rectangular cross-section is $3{,}10\text{ m}$ high and travels strictly within its lane (so that its point furthest from the center is at the outer edge of the lane, at $x = 3\text{ m}$). Algebraically determine the vertical clearance offered by the vault along that vertical line.
3. c) Determine whether the vehicle can travel along the lane without grazing the structure, and calculate the vertical safety margin at the most unfavorable point.

Find the equations of the tangent lines to the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$ that pass through the exterior point $P(0, 5)$.

Determine the equations of the tangent lines to the ellipse $x^2 + 2y^2 = 18$ that are parallel to the line with equation $x - y + 4 = 0$.

Calculate the values of the parameter $k$ such that the line $x + 2y + k = 0$ is tangent to the ellipse with equation $x^2 + 4y^2 = 20$.

Consider the circle with equation $x^2 + y^2 = 100$. With each point $A(x, y)$ on the circle, we associate a point $P(x, y')$ that has the same x-coordinate, but its y-coordinate is $60\%$ of the y-coordinate of $A$. Prove that the locus of the points $P$ is an ellipse and determine its equation, its centre and its foci.

A rectangular window is to be designed, inscribed in an elliptical opening with equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. The sides of the rectangle must be parallel to the axes of the ellipse. If the rectangle is required to have a total area of $24\text{ m}^2$, what should its exact dimensions be?

A main square is shaped like an ellipse. A straight path is to be paved connecting the leftmost focus, $F_1$, with a point $P$ on the ellipse located in the second quadrant. It is known that the major axis of the square measures $100\text{ m}$ and the minor axis measures $60\text{ m}$. If the angle formed by the segment $F_1P$ with the positive x-axis is $60^\circ$, calculate the exact length of the paved path.

An ellipse is centred at the origin and its axes of symmetry coincide with the Cartesian coordinate axes. Given that the ellipse passes through the points $A(2, \sqrt{3})$ and $B(\sqrt{2}, \frac{\sqrt{15}}{2})$, find the equation of the ellipse and calculate the exact distance between its foci.

Find the locus of the centres of the circles that are internally tangent to the circle with equation $x^2 + y^2 = 100$ and pass through the fixed point $A(6, 0)$. Prove that this locus is an ellipse, find its canonical equation and determine the coordinates of its foci.

To classify the relative position of a point $P(x_0, y_0)$ with respect to an ellipse (determining whether it is interior, exterior, or lies on the curve) whose foci are $F_1(-3, 0)$ and $F_2(3, 0)$ and with defining constant $2a = 10$, two students propose different procedures:

• Sofía's method (algebraic): Calculate $b$ from the fundamental relationship, construct the canonical equation $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, and compare the numerical value of $\dfrac{x_0^2}{a^2} + \dfrac{y_0^2}{b^2}$ with $1$.

• Carlos's method (metric): Use the definition as a locus by calculating the sum of Euclidean distances $d(P, F_1) + d(P, F_2)$ and comparing it directly with $2a = 10$.

1. a) Apply both methods in detail to the point $P\left(3, \dfrac{16}{5}\right)$, determine its relative position according to each one, and verify that both paths lead to the same conclusion.
2. b) Analyze the advantages and disadvantages of each method for this specific case and argue which one turns out to be faster and cleaner when working without a calculator.
3. c) Suppose that the foci were at the points not aligned with the axes $F_1(1, 2)$ and $F_2(5, 5)$, keeping the constant $2a = 10$. Justify with reasoning which of the two methods is clearly preferable in this new scenario, and explain the major algebraic complication that would arise when trying to apply the other one in 1.º de Bachillerato.

A marine environmental monitoring network has two acoustic reference buoys anchored at points $F_1(-6, 0)$ and $F_2(6, 0)$ (coordinates in kilometers). An underwater surveillance drone continuously travels along a fixed elliptical path centered at the origin with semi-major axis $a = 10\text{ km}$ and foci at these buoys.

To calibrate an interference sensor, the drone must be located at a point $P(x, y)$ on its elliptical path such that the distance to buoy $F_1$ is exactly three times the distance to buoy $F_2$; that is, $d(P, F_1) = 3\cdot d(P, F_2)$.

1. a) Without using the analytical coordinates of $P$ yet, deduce the exact numerical values of the distances $d(P, F_1)$ and $d(P, F_2)$ relying exclusively on the definition of the ellipse as a locus.
2. b) Write the canonical equation of the elliptical path and set up a system of equations to calculate the exact coordinates of all points $P$ in the plane that simultaneously satisfy the equation of the ellipse and the calibration condition.
3. c) Would it be possible to calibrate the sensor at a point on this same path if the required condition were $d(P, F_1) = 5\cdot d(P, F_2)$? Justify your answer using two independent geometric arguments: one based on the triangle inequality in the triangle with vertices $P$, $F_1$, and $F_2$, and another based on the minimum possible distance from any point on the ellipse to focus $F_2$.