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 centered 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 center 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 standard form.

Check whether the points $P(2, -3)$, $Q(5, 0)$, and $R(4, 2)$ are interior, exterior, or belong to the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$.

Find the equation of an ellipse centered at the origin, whose foci are 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 center and the length 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.

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

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 center of the river is $6\text{ m}$. Calculate at what height the arch is at a distance of $4\text{ m}$ from the center of the river.

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

Find the equation of the locus of the 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 until you 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 center of the room must each person be located 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 the 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 and calculate its center and the length of its semi-axes.

In a clockwork mechanism, a gear has an elliptical shape. It is known that its boundary equation is given by $x^2 + 4y^2 = 16$. It is desired to place a small pin 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 center of the gear.

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 $k$ so that the line $x + 2y + k = 0$ is tangent to the ellipse with equation $x^2 + 4y^2 = 20$.

Consider the circle with equation $x^2 + y^2 = 100$. To each point $A(x, y)$ on the circle is associated 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 center, and its foci.

A rectangular window is to be designed to be 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 to have a total area of $24\text{ m}^2$, what must its exact dimensions be?

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

An ellipse is centered at the origin and its axes of symmetry coincide with the Cartesian coordinate axes. If it is known 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 centers of the circles that are internally tangent to the circle with equation $x^2 + y^2 = 100$ and that pass through the fixed point $A(6, 0)$. Prove that this locus is an ellipse, find its standard equation, and determine the coordinates of its foci.