Exercises: Parabola

Find the standard equation of the parabola with vertex at the origin of coordinates and focus at the point $F(0, 3)$.

Determine the coordinates of the vertex, the focus, and the equation of the directrix of the parabola with equation $y^2 = -10x$.

Write the equation of the parabola whose vertex is the point $V(2, -1)$ and whose directrix is the line $x = 4$.

Find the locus of points in the plane that are equidistant from the point $A(0, 4)$ and the horizontal line $y = -4$.

Calculate the coordinates of the points of intersection of the parabola $x^2 = 4y$ with the bisector line of the first quadrant $y = x$.

Identify whether the general equation $y^2 - 4y - 6x + 10 = 0$ corresponds to a parabola and, if so, find the coordinates of its vertex.

The profile of a car halogen headlight has the shape of a parabola. If its equation relative to a frame of reference centered at the vertex is $y^2 = 12x$ (with measurements expressed in centimeters), at what distance from the vertex should the bulb be placed so that the light rays reflect parallel to the axis of symmetry?

Find the equation of the parabola with vertex at the origin, axis of symmetry coinciding with the x-axis, and passing through the point $P(-2, 4)$.

Given the general equation of the parabola $x^2 - 6x - 8y + 25 = 0$, complete the square to express it in its translated standard form and determine the coordinates of its vertex, its focus, and the equation of its directrix.

Find the equation of the tangent line to the parabola $y^2 = 8x$ at the point with y-coordinate $y = 4$.

A parabolic arch of a stone bridge has a span (width at the base) of $40\text{ m}$ and a maximum height of $10\text{ m}$. Calculate the height of the arch at a horizontal distance of $8\text{ m}$ from the center of the base.

Find the locus of the centers of the circles that are tangent to the line $y = -2$ and pass simultaneously through the point $P(0, 2)$. Identify the resulting conic section and detail its main elements.

The reflector of a parabolic antenna has an outer diameter of $2\text{ m}$ and a maximum depth of $0.25\text{ m}$. Determine at what distance from the vertex the focus is located where the signal receiver should be placed.

Determine the equation of the parabola with a vertical axis passing through the points $A(-1, 0)$, $B(1, -2)$ and $C(3, 4)$.

Study the relative position of the line $r: 2x + y - 3 = 0$ and the parabola $y^2 = 4x$. If they intersect, find the exact length of the chord determined by the intersection of the line and the parabola.

Determine the value of the parameter $k$ so that the line $y = 2x + k$ is tangent to the parabola $x^2 = 2y$.

Find the equations of the tangent lines to the parabola $y^2 = 4x$ drawn from the external point $P(-1, 0)$.

Using the definition of a parabola as a locus, find the general equation of the parabola with focus at point $F(2, 3)$ and directrix given by the oblique line $x + y - 1 = 0$.

A water cannon launches a jet whose trajectory describes a parabola. The water leaves from the origin of coordinates and the jet reaches a maximum height of $15\text{ m}$ at a horizontal distance of $20\text{ m}$. Determine analytically whether the jet of water will manage to pass over a wall $10\text{ m}$ high located at a horizontal distance of $30\text{ m}$ from the launch point.

Given the parabola with equation $y^2 = 2px$, prove analytically that the locus of the orthogonal projections of the focus onto the infinitely many tangent lines to the parabola is precisely the tangent line at the vertex.

Find the equation of the circle passing through the vertex of the parabola $y^2 = 8x$ and through the two points of intersection of this parabola with the vertical line $x = 2$.

Determine the coordinates of the point on the parabola $x^2 = 4y$ that is at the minimum distance from the line $r: x - y - 4 = 0$. Also, calculate this minimum distance.

Consider a parabolic mirror whose cross-section corresponds to the equation $y^2 = 4x$. A ray of light, parallel to the x-axis, strikes the mirror at point $P(1, 2)$. Prove analytically that the line containing the reflected ray passes precisely through the focus of the parabola.

A parabola has its vertex at point $V(1, 2)$ and its axis of symmetry is parallel to the y-axis. Given that the parabola is tangent to the line $y = x - 1$, find the general equation of the parabola and the exact coordinates of its focus.